A nonlinear tensor field and the second metric tensor

In our preceding paper {see [L. Sh. Grigorian and S. Gottlöber, Astrofizika (in press)]} we investigated a self-gravitating system consisting of a scalar field and a linear tensor field _ik= _ki with minimal coupling and with allowance for the action of vacuum polarization effects. In the present paper we investigate the case of a nonlinear tensor field _ik. The action S ⁽⁾ of the field _ik is determined by the difference R_ik — _ik, where R_ik is the space-time Ricci tensor and Rik is the analogous quantity constructed using the metric _ik=g_ik+_ik induced by _ik ( is a free parameter). Here S ⁽⁾ coincides with the previously known expression for the action of a linear field _ik. Equations of motion are derived for _ik in curved space-time. The energy-momentum metric tensor, determining the contribution of _ik to the gravitational field equations, is calculated.Translated from Astrofizika, Vol. 39, No. 1, pp. 135–144, January-March, 1996.     

Pulsar radio emission from expanding charge sheets

We semi-quantitatively calculate the distribution of energy in frequency and angle emitted from a sheet of charges that are moving out relativistically along dipolar magnetic field lines originating near the magnetic polar caps of a rotating neutron star. The angular distribution is conical with the angle of maximum intensity varying with frequency as ^–1/4 for _c 2 _c /(R _M ²), whereR_M is the initial angular radius of the charge sheet at the surface of the star of radiusR. At higher frequencies the width of the angular cone remains constant. The radiation is linearly polarized with the polarization vector in the plane of the line of sight and the magnetic axis. A sheet of uniform charge density and finite thickness has a frequency spectrum that varies from ^–3/2 to ^–4 for _c and _c , respectively. These features are in good general agreement with the observed characteristics of the intensity, pulse shape, and frequency spectrum of the radio pulses from pulsars.Operated by Associated Universities, Inc., under contract with the National Science Foundation.     

Compton scattering of relativistic electrons in compact X-ray sources

The problem of single Compton scattering is considered and the resulting spectrum, angular distribution and polarization of scattered photons in a general case are obtained. The inverse Compton scattering (ICS) for arbitrary energies of electronsE and photons ₀ is investigated in detail. In the case of isotropically-distributed initial photons and relativistic electrons, a strong rise of the scattered spectrum near the upper edge takes place, starting from the values of the characteristic parameterb4E ₀10 (in units of mc²=1). The energy-loss rate of relativistic electrons due to ICS is calculated. It is shown that the relativistic electrons of the energiesE100 MeV, when scattering on the X-rays with ₀~10KeV, transmit the dominant part of their energy to the photons which fall after scattering into the energy range of the electrons (100 MeV).The radiation spectrum of ICS, as well as the energy-losses of relativistic electrons distributed by power-lawE ^–, are calculated. The radiation spectrum reveals the power-law behaviour with the different indices in two limits: the dependence ^–(1)/2 at ₀1 gradually changes to ^–(+1) ln (₀) law for ₀1.     

Некоторые асимптотические решения в ограниченной задаче трех тел

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Some asymptotic solutions in the restricted problem of three bodies by L. G. Lukjanov.

Some particular solutions of the plane restricted problem of three bodies in the form of Liapunov's series are obtained. These solutions asymptotically approach the Lagrange solutions. Convergence is proved.

     

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 .     

A Simple Model for the Solar Cycle

A simple nonlinear model which describes the 11-year solar cycle can be derived from the usual – dynamo theory in the form of a Van der Pol equation. Solar activity displays also small-scale inter-cycle persistent stochastic oscillations with a Hurst exponent of the order of H0.76±0.01. The results obtained from the Van der Pol oscillator superimposed on a fractional Brownian motion which describes the stochastic fluctuations are presented.     

A Razin-Tsytovich effect for bremsstrahlung

It is shown that bremsstrahlung from electrons with Lorentz factor 1 is suppressed for >_p in a plasma with plasma frequency _p compared with emission in vacuo. For _p the ratio of the power emitted per unit frequency in the plasma to that in vacuo varies as ².This suppression effect is analogous to the suppression of synchrotron radiation in a plasma (Razin-Tsytovich effect). It is argued that such suppression is a characteristic property of emission by relativistic particles in a plasma.     

Vertical motions in an intense magnetic flux tube

The recent discovery of localised intense magnetic fields in the solar photosphere is one of the major surprises of the past few years. Here we consider the theoretical nature of small amplitude motions in such an intense magnetic flux tube, within which the field strength may reach 2 kG. We give a systematic derivation of the governing expansion equations for a vertical, slender tube, taking into account the dependence upon height of the buoyancy, compressibility and magnetic forces. Several special cases (e.g., the isothermal atmosphere) are considered as well as a more realistic, non-isothermal, solar atmosphere. The expansion procedure is shown to give good results in the special case of a uniform basic-state (in which gravity is negligible) and for which a more exact treatment is possible.The form of both pressure and velocity perturbations within the tube is discussed. The nature of pressure perturbations depends upon a critical transition frequency, _p , which in turn is dependent upon depth, field strength, pressure and density in the basic (unperturbed) state of the tube. At a given depth in the tube pressure oscillations are possible only for frequencies greater than _p for frequencies below _p exponentially decaying (evanescent) pressure modes occur. In a similar fashion the nature of motions within the flux tube depends upon a transition frequency, _v . At a given depth within the tube vertically propagating waves are possible only for frequencies greater than _v ; for frequencies below _v exponentially decaying (evanscent) motions occur.The dependence of both _v and _p on depth is determined for each of the special cases, and for a realistic solar atmosphere. It is found that the use of an isothermal atmosphere, instead of a more realistic temperature profile, may well give misleading results.For the solar atmosphere it is found that _v is zero at about 12 km above optical depth ₅₀₀₀= 1, thereafter rising to a maximum of 0.04 s^–1 at some 600 km above ₅₀₀₀ = 1. Below ₅₀₀₀ = 1, in the convection zone, _v has a maximum of 0.013 s^–1. The transition frequency, _p , for the pressure perturbations, is peaked at 0.1 s^–1 just below ₅₀₀₀ = 1, falling to a minimum of 0.02 s^–1 at about one scale-height deeper in the tube     

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 .     

Stability of the triangular libration points of the circular restricted problem in the presence of resonances

The stability of triangular libration points, when the bigger primary is a source of radiation and the smaller primary is an oblate spheroid. has been investigated in the resonance cases ₁ = 2₂ and ₁ = 3₂. The motion is unstable for all the values of parameters q and A when ₁ = 2₂ and the motion is unstable and stable depending upon the values of the parameters q and A when ₁ = 3₂. Here q is the radiation parameter and A is the oblateness parameter.     

Cyclotron waves in the solar wind

Cyclotron waves in the solar wind near 1 AU with frequencies well below the electron cyclotron frequency and wavelengths much larger than the electron cyclotron radius but less than the proton cyclotron radius are considered. The cyclotron radii are defined from parallel thermal velocity of electron component and proton component with respect to the interplanetary magnetic field. No LH cyclotron waves are found to propagate for _p < 0, where _p 1 –T _p/T _p is the temperature anisotropy of the proton component with respect to the interplanetary magnetic field. The damping or growth of RH cyclotron waves is found to depend on the frequency range and the temperature anisotropy of the proton component. The RH cyclotron waves are damped in the frequency range _r | _p | _p for _p < 0, where _p is the proton cyclotron frequency. RH cyclotron instabilities occur in the frequency range | _p | _p > _r > | _p | _p /(1– _r ) for _p < 0. The marginal state is at _r =| _p | _p .Abstract presented at theInternational Symposium on Solar-Terrestrial, São Paulo, Brazil, 17–22 June, 1974     

The 4∶1 resonance

We study the resonance ₁₂ = 41 and some near-resonance cases. The main peculiarity of this resonance is that for ₁₂ < 4 the characteristic of the central periodic orbits is broken into two and each part is joined with a resonant characteristic. This behaviour is described theoretically by means of the third integral. It seems that there are infinite families of simple periodic orbits near the escape region. Finally, a comparison is made with the cases near the ₁₂ = 21 resonance.     

High-frequency radiation from a particle falling into a Kerr black hole

The high-frequency electromagnetic and gravitational radiation from a relativistic particle falling into a Kerr and Schwarzschild black hole is considered. The spectral and angular distributions of the radiation power are calculated by the WKB technique to Teukolsky's equations. The spectra obtained have a characteristic exponential cut-off at the frequency = _char. which is proportional to the particle Lorentz factor =(1–v ²/c²)^–1/2. At the frequencies as low as those compared with _char. both electromagnetic and gravitational spectra are flat. The amount of the energy emitted in the low-frequency modes of the radiation depends strongly on the radiation spin. It is proportional to ln for the electromagnetic and to ³ for the gravitational radiation.     

Systemes hamiltoniens au voisinage d'une solution d'equilibre. II:Stabilité des solutions périodiques

Resume On étudie la stabilité des solutions périodiques d'un couplage de systèmes linéaires au voisinage de résonances. Les valeurs propres distinctes _k de la matrice du système linéaire non perturbé sont telles que _k–_j=iq pour tout couple [k, j]; i=–1, q est un nombre entier, la fréquence de la solution. Une application est faite pour un système à trois degrés de liberté au voisinage de la résonance 221.

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Hamiltonian systems in the neighbourhood of an equilibrium solution. II:The stability of periodic solutions

The stability of the periodic solutions for an Hamiltonian system is investigated. Here the distinct eigen values _k of the matrix of the unperturbated linear system are such that _k–_j=iq for any [k, j]; i=–1, q is an integer, is the frequency of the periodic solution. An application is made for a system with three degrees of freedom, near the resonance 221.

     

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.     

On the origin of type III fundamental

A two-component scheme for the generation of type III fundamental radiation is proposed. The first component of the fundamental arises at a plasma level _{L t} because of the Rayleigh scattering of the plasma waves into electromagnetic radiation. The other component arises at _{L t} /2 because of the decay of the first component into plasma waves and the subsequent rescattering of the plasma waves into electromagnetic radiation _t 2( _t /2). By its properties (location, directivity, polarization) the second component is essentially the same as the second harmonic radiation produced by a stream of fast electrons at _L ( _t /2). This scheme is used to solve the main problems (localization and directivity of the source, polarization of type III fundamental) of the harmonic theory of type III solar bursts.     

Liminosity distribution function for visual binary stars

The cosmic scatter of 147 of the best known visual binary stars on the Main Sequence is discussed and a new estimation of the luminosity distribution function for multiple star systems is presented. As long as the mass ratioq of a close binary is not smaller than 0.5, the distribution of close binary components is identical to the van Rhijn liminosity-function. For smaller mass ratios (q<0.5) the number of close companions decreases rapidly. It appears that less than 13% of visual binaries in our sample are simple binary systems.Communication presented at the International Conference on Astrometric Binaries, held on 13–15 June, 1984, at the Remeis-Sternwarte Bamberg, Germany, to commemorate the 200th anniversary of the birth of Friedrich Wilhelm Bessel (1784–1846).     

Models of pulsating low-massive yellow supergiants

The nonlinear self-excited oscillations of the envelopes of low-massive highly luminous stars are described. The parameters for these models wereM=0.8M ,M _bol=–5.5, –5.84 mag,T _eff=4500, 5000, 5500 K. The oscillations have been found to consist of the standing wave pulsation near the envelope bottom and running waves in outer layers. The ratio of the standing wave frequency _s to the average frequency of the running waves _r increases with the stellar luminosity: _s / _r =1.7 whenM _bol=–5.5 mag and _s / _r =2.4 whenM _bol=–5.84 mag. The frequency of oscillations near the photosphere is found to be in close agreement with the critical frequency for running waves. Mass loss from these stars is caused by shocks. It has been shown that agreement between FG Sge's period change observed during the last decade and the period-luminosity relation for double shell stars takes place when FG Sge's luminosity isM _bol=–5.96 mag.     

Multiple expansion of the tidal potential

The Earth tidal deformation causes an additional gravitational potential. Its effect on the Moon orbital motion has been studied by several authors.In this contribution, we develop this additional potential without specifying the inertial frame chosen.For this purpose, we use the properties of the representation of rotation groups in 3 dimensions space. We finally obtain the interaction potential between the distorted Earth and the Moon which is a necessary preliminary to the study of the evolution of the Earth-Moon system.Nomenclature T.R.O Tide raising object - (, , ) Spherical coordinates of the T.R.O. - (J, _E ) Earth spin axis orientation. _E is the longitude of the ascending node of Earth's equator on thexy-plane - (a ,I ,e , , ,M ) Elliptics elements of the T.R.O     

Quantum effects in cyclotron plasma absorption

It is shown that X-ray radiation of neutron stars with magnetic fieldsB=10¹¹–10¹³ G near cyclotron resonances=s _B (s=1,2,...) is deeply affected by such quantum effects as electron-positron vacuum polarization (significant at V=3×10²⁸ n _e ^–1 (B/B _C ⁴)1, whereB _C =4.4×10¹³G), the quantizing character of the magnetic field (significant atV=3 x 10²⁸ n _e ^–1 (B/B _c)⁴1 whereB _c =4.4 x 10¹³G), the non-harmonic character of the Landau levels, and the quantum recoil of electrons. The latter two factors shift the resonances by the frequency –s ² _B (B/2B _c )sin², being the angle between the direction of radiation propagation and the magnetic field. IfVV ₀ (for 1,V ₀^–1=(mc ²/2T)^1/2), the normal mode (NM) polarizations, as well as the absorption coefficientk ₁ of the extraordinary NM in the Doppler core of the first resonance (|–| _B cos ), is only slightly affected by varyingb and/orV, whereas for the ordinary NM (at 1)k ₂k ₁ ²[b + (3 + tan²–2V)²]k ₁. For sufficiently largeb and/orV the quantum effects amplify resonant absorption of the ordinary NM at _B , with spin-flip transitions playing a major role atb1+V ². IfVV ₀, the coefficientsk ₁ andk ₂ in the Doppler core of the resonance are of the same order and acquire some peculiar features (shifts, intersections, etc.), with the NM polarizations depending sharply on and being strongly non-orthogonal. AtVV ₀,k ₂=k ₁(cos² +B/2B _C ) and the polarizations are almost linear. Near high resonances (s2), as a rule,k _1,2(1 + b) ^{s–1 2s–3} i.e., absorption increases withb due to replacement of the thermal energy of the transverse motion of electron,T, by the magnetic energy _B . The above effects should be taken into account for an interpretation of observational data on X-ray pulsars (e.g., Her X-1) and other X-ray sources associated with neutron stars.