On the figure of the planet Mercury

The figure of Mercury is estimated in terms of an isostatic form of equilibrium which tends to be controlled by the situation near perihelion passage at the 32 resonance spin rate. The ratios of the principal moments of inertia for Mercury are: (1)(C–A)/C7×10^–5; (2)(C–B)/C5×10^–5 and (3)(B–A)/C2×10^–5. The thermal effect on Mercury's figure during solidification forces Mercury's rotation to be trapped in the 32 resonance lock as its spin rate is being slowed by tidal effects. It is shown that the process of trapping of Mercury has been naturally affected by the instantaneous solidification of Mercury into a shape with two thermal bulges, and that the two permanent thermal bulges stabilize the planet's rotation.     

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.     

On the rotation of rigid Venus

According to A.A. Khentov Venus' rotation is in the quasi-stationary state as a result of the balance interaction of the solar tidal torque with the aerodynamical torque of the rotating Venus' atmosphere. In case of the nonconservative forces are negligible and the solar attraction is the stabilizing factor, the rotation of the rigid Venus may be assumed as the first approximation. The theory of the rotation of the rigid Venus in the coordinates,, had been constructed. It have been found that Venus rotates almost uniformly and the libration harmonics are negligible.     

PO — Prospects in interstellar medium

Proton transfer and condensation reactions followed by dissociative electron recombination and charge transfer processes may yield a significant concentration of PO in interstellar clouds. Relevant terrestrial PO lines in the ultraviolet, infrared and microwave regions are listed in order to aid the detection of PO in interstellar clouds. Within the framework of Klein-Dunham potential, spontaneous emission rates and oscillator strengths of strong (v=0 sequence) beta bands of the PO molecule are estimated. For SgrB2 source, under optically thin case, each of the two strongest²_1/2,J=2.5–1.5 rotational transitions (108.998 GHz and 109.206 GHz) of PO would have antenna temperature less than 125 mk for beam efficiency0.6 and =0.7D (electric dipole moment of PO in thev=0 level of theX ² state). For Orion molecular cloud, PO (4, 4.5, 5-3, 3.5, 4) (e, f) lines are excitable at 196.305 and 196.500 GHz and each line would have antenna temperature less than 110 mK.This work was partially supported by Fapesp and CNPq, Brazil, under contract numbers 80/1659-0 and 30.4076/77-FA01, respectively.     

On a Babcock-Leighton dynamo model with a deep-seated generating layer for the toroidal magnetic field, II

In a previous paper (Paper I), we studied a dynamo model of the Babcock-Leighton type (i.e., the surface eruptions of toroidal magnetic field are the source for the poloidal field) that included a thin, deep seated, generating layer (GL) for the toroidal field, B. Meridional motions (of the order of 12 m s^–1 at the surface), rising at the equator and sinking at the poles were essential for the dynamo action. The induction equation was solved by approximating the latitudinal dependence of the fields by Legendre polynomials. No solutions were found with _p = _f where _p and _f are the fluxes for the preceding and following spot, respectively. The solutions presented in Paper I, had _p = –0.5 _f , were oscillatory in time, and large radial fields, B, were present at the surface.Here, we resume the study of Paper I with a different numerical approach allowing for a much higher resolution in , the polar angle. The time dependent partial differential equations for the toroidal and poloidal field are solved with the help of a second order, time and space centered, finite difference scheme. Oscillatory solutions with _p = _f are found for various values of the meridional motions and diffusivity coefficients. The surface values of B, while considerably smaller than those of Paper I, are still unacceptably large, specially at the poles. The reason can be traced to the eruption of toroidal field at high latitudes. It appears that in order to obtain small values for the radial field in the polar regions, high latitude sources ( smaller than /4, say), must reach their maximum below the surface. Weaker meridional motions near the poles than in the equatorial region are also suggested.     

Unexplained empirical relations among certain scatterings

When the differential cross-section data of ⁺ -proton scattering are compared to the data of proton-alpha scattering at momenta related by 4p₄=p (where p₄ is the momentum of p-alpha scattering and p is the momentum of ⁺ -proton scattering) a correlation is observed between p=125 MeV/c and 1100 MeV/c. The correlation is most pronounced in the region of the lowest energy resonance of both scatterings (p210 MeV/c). A less pronounced correlation (in the lower energy regions) among four scatterings is observed when the differential cross-section data of ⁺ -p and p-alpha scatterings and the data of p-³He and protondeuteron scatterings are compared using relations 4p₄=3p₃=2p₂=p (where p₃ is the p-³He momentum and p₂ is the proton-deuteron momentum). The facts strongly suggest something of physical significance but no theoretical explanation is known. Indeed, an explanation seems to be beyond the scope of existing theory.     

Hydrodynamic models for population-II cepheids

The nonlinear self-excited pulsations of population-II stars with mass 0.6M and luminosities from 128 to 1280L are studied. The pulsation periods are found to be in the range of 1.3 to 19 days. An increase of the stellar luminosity is shown to be accompanied by an increasing nonadiabaticity and decreasing efficiency of the radiative damping region. This leads to both an increase of the growth rate while pulsations are exciting and an increase of the oscillation amplitude of the limit cycle. In the models withL800L the efficiency of the radiative damping region becomes so small that amplitude growth ceases due to a dissipation of the mechanical energy by shocks in the stellar atmosphere. The models with periods of from 1.3 to 3 days show the bump on their light curves. The bump is connected with a travelling pulse generated at the antinode of the second overtone at maximum compression. The time delays estimated for the pulses reflected of the stellar core are in a good agreement with the pulse resonance condition proposed by Aikawa and Whitney (1983). The model with the period of 2.1 days revealed double resonance ₀ = 2₂, 2₀ = 3₁ causing alternating oscillations with slightly different periods and amplitudes. The models with period of 10 days and longer reveal the resonance ₀ = 2₁. This resonance causes the flat top on the light curve at a period of about 10 days and appearance of a shallow alternating minimum at longer periods, as is observed in RV Tau variables. The theoretical period-luminosity relation proposed for population-II cepheids is in good agreement with that obtained from observations.     

Dynamics and internal structure of an eruptive prominence

The kinematics and the development of the internal structure in the eruptive prominence of August 16, 1988 are described. The prominence exposed helical structure, and the pitch of the fine structure filaments was measured. The evolution of the pitch was measured in the legs of the prominence and at its summit from the pre-eruptive phase up to the late phases of the eruption. The pitch angle was decreasing in the legs as well as at the summit. However, the observations indicate that the integral twist remained constant. The prominence was twisted more at the summit where it was wider than in the legs. The effective twist at the prominence summit was approximately 20 and in the legs it amounted to about 8 . Such a ratio did not change during the eruption, i.e., no redistribution of the twist was observed within the accuracy of measurements. The nature of the instability causing the eruption is discussed and the energetics of the process is considered.     

Effect of drift-resonance broadening on radial diffusion in the magnetosphere

In the quasilinear theory of magnetospheric radial diffusion caused by fluctuating electrostatic (E) or magnetic (B) fields, the diffusion coefficientD _LLis proportional to the spectral density of E or B at the particle drift frequency ₃/2. Since ₃ varies withL at fixedM andJ (adiabatic invariants), the drift resonance =₃ can be maintained only transiently, and therefore is not perfectly sharp. Its bandwidth ^* is approximately (16D _LL /L ²₃)^1/3₃. In magnetospheric radial diffusion caused mainly by electrostatic fluctuations, the value of ^*₃ typically exceeds 0.4 for particle energiesE40 keV. However, the numerical value ofD _LLis correctly given (within 1% in all cases) by quasilinear theory because the spectrum of E is rather flat at resonance frequencies for which the bandwidth is an appreciable fraction of ₃. (Numerical conclusions are based on a quasilinear model forD _LLused successfully by Cornwall in 1972.)     

The lunar mean moment of inertia and the size of the Moon's core

In this paper the relation between the uncertainty of the Moon's mean moment of inertia (I/Ma ²) and that of the core density _c is discussed with a two-layer model of the Moon - a mantle obeying Roche's law of the density distribution and a homogeneous core (Fe-core or Fe-FeS-core). When the uncertainty of I/Ma ² is 0.0023 (that is the accuracy in present observation), a core with radius of 450 km will be appropriate to the limitation of _c about 1 g cm^–3. Considering the accuracy obtained in space explorations, and the compressibility and the quasi-homogeneity of the Moon, we suggest that the parameters C ₂₀, , , a, and GM of the Moon should define as primary constants, but C ₂₂ and C/Ma ² as derived constants. Therefore, the ratio of mass of Moon to that of Earth in the IAU (1976) system of astronomical constants will become a deducible constant.     

On the interpretation of the π-component splitting in sunspot spectra

It is shown that in order to explain the observed splitting of the -component in the sunspot umbra spectrum by the hypothesis of the coexistence in sunspots of weak- and strong-field regions with opposite polarities, one has to admit the additional assumption that in the weak-field regions the Doppler halfwidth ( _D) and the ratio between line opacity and continuum opacity ( ₀) are both less than those in the strong-field regions.     

Asymmetric librations in exterior resonances

The purpose of this paper is to present a general analysis of the planar circular restricted problem of three bodies in the case of exterior mean-motion resonances. Particularly, our aim is to map the phase space of various commensurabilities and determine the singular solutions of the averaged system, comparing them to the well-known case of interior resonances.In some commensurabilities (e.g. 1/2, 1/3) we show the existence of asymmetric librations; that is, librations in which the stationary value of the critical angle =(p+q)₁–p–q is not equal to either zero or . The origin, stability and morphogenesis of these solutions are discussed and compared to symmetric librations. However, in some other resonances (e.g. 2/3, 3/4), these fixed points of the mean system seem to be absent. Librations in such cases are restricted to =0 mod(). Asymmetric singular solutions of the planar circular problem are unkown in the case of interior resonances and cannot be reproduced by the reduced Andoyer Hamiltonian known as the Second Fundamental Model for Resonance. However, we show that the extended version of this Hamiltonian function, in which harmonics up to order two are considered, can reproduce fairly well the principal topological characteristics of the phase space and thereby constitutes a simple and useful analytical approximation for these resonances.     

Orientation and resonance locks for satellites in the elliptic orbit

In order to achieve the maximum strength of higher resonance locks for satellites in the elliptic orbit, the condition of satellite orientation during the process of deployment is established. It is shown that for maximum strength locks the axis of the minimum moment of inertia of satellites should point toward the attracting body at ±(5/8) and 0 values of the true anomalyf. This condition of deployment is applicable to all cases of resonance rotation regardless of the value of lock numberk and orbit eccentricitye.     

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 .     

On the motion of a satellite in resonance with its rotating planet

The influence of resonance perturbations due to the gravitational field of an oblate planet on its satellite whose motion is commensurable with rotation of the planet has been investigated. It has been shown that in special case of the critical inclination or circular orbit the Lagrange equations can be integrated for all resonance terms simultaneously. The method is applied to the investigation of the motion of the 12-hour communication and navigation satellites of the Molniya and Navstar type. The computations has been performed by the use of four models of the geopotential.     

Fourier analysis of the light curves of eclipsing variables-XXV: Error analysis in the frequency-domain

The aim of the present paper has been two-fold. In the first part (Sections 1–2), closed algebraic formulae will be set up furnishing the momentsA of the light curves of arbitrary index , and, due to arbitrary type of eclipses, in terms of the coefficientsa of Fourier cosine series obtained by least-squares fit to the given data; and the uncertainty of the momentsA deduced from that of thea 's.In the second part (Sections 3–4) we shall establish the explicit forms of the lincar functions r _1,2, (cosi) and L ₁ for the variation of the respective elements expressible likewise in terms of the Fourier coefficientsa . The probable errors of these elements can then be identified with those of the respective linear functions, and are obtainable from the same matrix of coefficients which furnished the most probable values of the elements.     

Linearly expanding universes in JBD-theory

The assumption of a linearly expanding universe for the JBD-cosmological equations generates a set of solutions for the barotropic equations of statep= (=const.). These solutions turn out to be valid for closed space-except in the casep= which is for open space. The gravitational constant which is inversely proportional to the scalar field increases with time if >–1/3 and decreases for <–1/3. No solution exists for =1/3. The Brans-Dicke parameter is negative if <–1/3.     

Two-dimensional stochastic motions and the problem of differential rotation

This paper deals with the spatial dependence of the angular velocity in a rotating turbulent fluid sphere. The original turbulence unaffected by the global rotation is assumed to be two-dimensional where the stochastic force field producing the turbulence does not possess a radial component. By using results of earlier papers we proceed to the treatment of a rotational rate, , no longer small compared to _c (frequency of turbulent mode). It is shown that for _c the angular velocity increases with increasing radius but no latitudinal dependence exists. Contrary to this, for 2 _c an equatorial acceleration is possible and related to negativity of the two-dimensional eddy viscosity. Furthermore, the outer layers rotate faster than the inner ones. These findings coincide with Gilman's numerical results. Ward's observations, as well as the characteristic scales of supergranulation and giant cells, suggest the presence of negative two-dimensional eddy viscosity on the Sun.     

The role of loop footpoints rotation in single loop flares

We consider that single loop flares can be caused by the rotation of loop footpoints. Choosing a typical geometry for this case we find from MHD equations self-consistent expressions and a set equations governing behaviour of all physical quantities. Numerical simulations have revealed that under the determined conditions for the initial azimuthal velocity and current the pinch instability takes place. The most important parameters of the problem are the plasma and the ratio of the initial values of longitudinal and poloidal components of the magnetic field-B ₁. Thus, calculations show that the critical pinch time increases with the increase ofB ₁ and decreases with the increase of plasma . So the most effective flares are probable for the most high loops with strong currents. ForB ₁=10 and =0.01 the critical pinch time is 2.5 s. The critical twist angle for magnetic field depends on the initial one. For low intial twist which corresponds to bigB ₁ the critical one is more less. For exampleB ₁=30 gives 1.8 (when ratio of loop length and radius is 10). Geometrical analysis shows that the present model can explain (for high photospheric rotation) single loop flares taking place on different parts of the loop as on the top of it as closer to one of the footpoints. It depends on the relative rotation momentum of loop footpoints. Subject headings: MHD-Sun:flares.