Theory of the Trojan asteroids,IV

In a previous publication (1977) the author has constructed a family () of long-periodic orbits in the Trojan case of the restricted problems of three bodies. Here he constructs the domain of the analytical solution of the problem of the motion, excluding the vicinity of thecritical divisor which vanishes at the exact commensurability of the natural frequencies ₁ and ₂. In terms of thecritical masses m_j(₂), or the associatedcritical energies _j ² (m), is the intersection of the intervals ofshallow resonance, of the form. Inasmuch as the intervals |₂– _j ² |<_j ofdeep resonance aredisjoint, it follows that (1) the disjointed family () embraces the tadpole branch, 0²1, lying in: and (2) despite the clustering of _j ² (m) atj=, the family () includes, for ²=1, an asymptoticseparatrix that terminates the branch in the vicinity of the Lagrangian pointL ₃.In a similar manner, the family () can be extended to the horseshoe branch 1<_{2 2} ² .     

On the stability of equilibria of Hamiltonian systems near the main resonances

The equilibrium point O of an autonomous Hamiltonian system of two degrees of freedom is considered for small-oscillation frequencies related as ₂=2₁+. If under the precise resonance (=0) the equilibrium is unstable, the inner diameter () of the domain of stability containing the point O is estimated. It is shown that for the normalized variables ()/b where b is the corresponding resonance coefficient. The estimates () for other main resonances are reported.     

The application of a non-static spherically-symmetric metric to clusters of galaxies

An approximate metric is found which represents a sphere of matter embedded in a background of dust. The use of this metric in conjunction with the Friedmann equations gives values of for the three possible values ofk as +6×10^–36 (k=+1), +3×10^–35 (k=0), +10^–36 (k=–1). These values depend on data regarding clusters of galaxies, and are probably accurate to within an order of magnitude given the correctness of the assumptions on which their derivation rests.     

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.     

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.     

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     

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.     

Космическое черенковское излучение

, . N/E, E _r, (E _r)=1. , (, ) . .     

The motion of a satellite in an axi-symmetric gravitational field

The equations for the variation of the osculating elements of a satellite moving in an axi-symmetric gravitational field are integrated to yield the complete first-order perturbations for the elements of the orbit. The expressions obtained include the effects produced by the second to eighth spherical harmonics. The orbital elements are presented in the most general form of summations by means of Hansen coefficients. Due to their general forms it is a simple matter to estimate the perturbations of any higher harmonic by simply increasing the index of summation. Finally, this paper gives the respective general expressions for the secular perturbations of the orbital elements. The formulae presented should be useful for the reductions of Earth-satellite observations and geopotential studies based on them.List of Symbols semi-major axis - C _jk ⁿ (, ) cosine functions of and - e eccentricity of the orbit - f acceleration vector of perturbing force - f sin²t - i inclination of the orbit - J _n coefficients in the potential expansion - M mean anomaly - n mean motion - p semi-latus rectum of the orbit - R, S, andW components of the perturbing acceleration - r radius-vector of satellite - r magnitude ofr - S _jk ⁿ (, ) sine functions of and - T time of perigee passage - u argument of latitude - U gravitational potential - true anomaly - V perturbing potential - G(M₊+m) (gravitational constant times the sum of the masses of Earth and satellite) - _n,k coefficients ofR component of disturbing acceleration (funtions off) - _n,k coefficients ofS andW components of disturbing acceleration (functions off) - mean anomaly at timet=0 - X ₀ ^n,m zero-order Hansen coefficients - argument of perigee - right ascension of the ascending node     

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.     

A search for discrete gamma-ray sources of energy greater than 2×1012 eV

The results of the observations to search gamma-ray sources with the energy greater than 2×10¹² eV, which were made in Crimean Astrophysical Observatory during the years 1969–73 are presented. A technique of the detection of the EAS Cerenkov flashes was used.The quality of the data obtained is analysed. The criteria for the selection of the data free from meteorological variations are considered.It was shown that two objects, namely, Cyg X-3 and Cas -1, may be the sources of high-energy gamma quanta. It is probable that the object with the coordinates =05^h15^m, =+1° is the source of gamma-rays as well. An unidentified object Cas -1 is variable: gamma-ray flux was observed twice — in Sepember–October 1971 and in December 1972. It is possible that the flux from Cyg X-3 has a period of 4.8 hr.

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I I , I I , - >2.10¹² . I . I , I I, I ., - -1 Cyg -3- -I . , =0515 ·=+1° -.I -1 I: I J I- - 1971 1972 . Cyg -3, , - T=4.8 .

     

Гидромагнитное Излучение Межпланетной Плазмы

0–2 ( ). . . .     

The effect of chemical abundance oscillations on the pulsational properties of A 5M ⊙ hydrogen-helium star

A model of a first generation intermediate star of 5M , with Z=0 has been considered. The model is at an advanced stage of its evolution and has a double shell burning. It burns helium in the inner shell, and hydrogen, via CNO cycle, in the outer shell. =(log/log) _T and _T =(log/logT) were computed allowing for the oscillations of the relative mass abundance of the reagents in nuclear reactions. Including =(log/log) _T and =(log/logT) of mean molecular weight and the effect of the oscillations of abundances due to nuclear reactions, stability was studied. Contrary to the results of the static calculations, we found that instability due to the excitation mechanism provided by the high temperature sensitivity of energy generation rate propagates up to the surface. Thus the model in question was found to be unstable against radial adiabatic pulsations, in its fundamental mode.     

Far-infrared emission from an intergalactic dust cloud?

We used theIRAS All Sky Maps in order to search for infrared emission in the direction of the Okroy Cloud (R.A.=12^h50^m, =22°). An enhancement of 100 m diffuse emission is evident in such a region, with an anomalous value of the ratioI _v (100 m)/A _v ; hydrogen 21 cm emission is also present with low radial speed, thus suggesting that the cloud could be a satellite of our Galaxy.     

Models of clusters of point masses with large central redshifts

Conclusions In the Newtonian case we have obtained an isotropic self-consistent distribution of gravitationally interacting point masses which satisfies the transport equation without collisions, and the gravitational equation for an arbitrary powerfunction density distribution =r^–s, s<3.For =r^–2 the analogous self-consistent solution was obtained for the anisotropic distribution function both in Newtonian and GTR cases.The GTR solutions with =r^–2 have central redshifts which increase without limit in accordance with the law 1+zr^–1/ as we approach the center. In the isotropic case, they appear to be stable when the mean velocities are much less than the velocity of light u<0.2c, >21.The hydrodynamic GTR solution was found for a perfect gas at constant temperature (but variable T=T(g₀₀)^1/2) which also has z for r0.We should like to thank K. Thorne, L. Hazin, and M. Podurets for valuable discussions. K. Thorne was particularly helpful in supplying unpublished results on circular orbits obtained by American authors.Astrofizika, Vol. 5, No. 2, pp. 223–234, 1969     

Оπорная селенодезическая система координат по гелиометрическим наБлюдениям луны В казани

, 1969–1975 .., 32 . , .     

Approximate expressions for Gyrosynchrotron radiation in Transverse Propagation

Two sets of accurate approximate expressions for the gyrosynchrotron radiation in the transverse propagation case are presented for the first time. They contain emissivity _–/BNand absorptivity _– B/Nfor e-mode, effective temperature T _effand frequency of peak brightness _p. The expressions are designed for the range 2 to 7 of electron energy spectral index and for the ranges from 2 to 10 and 10 to 100 of harmonic numbers s(=/_B). Their statistical error is, respectively, ±18% and ±29% for _–/BNand _– B/Nfor 10/_B100, ±128% and and ±170% for 2/_B10.     

Annihilation of antimatter meteors

Antimatter meteors probably enter the Earth's atmosphere. If they have the ability to escape complete vaporization during their infall flight, it may be possible, that a fraction of their original mass could survive for short or long time, depending on the mechanisms of ablation. In case of ablation through the annihilation process only, the lifetime of such an object is following the simple relation = (N _L R)/(rA), where andA are the density and the atomic weight of the antimatter fragment respectively,R is its radius,r is the rate of annihilation per cm² of its surface, and N _L is the Loschmidt number.     

General-relativistic hierarchical cosmology: An exact model

An exact solution of Einstein's equation is stated in which the density (), pressure (p), scale factorS and metric coefficients are functions of only one dimensionless self-similar variable,ct/R, wheret is cosmic time andR is a co-moving radial coordinate. The solution represents a cosmology that begins as a static sphere having R ^–2 and evolves into an expanding model which is pressure-free and has a hierarchical type of density law ( R ^–, approximately, with =a number, 02). It is suggested that this model should supersede the previous models of Wesson and other workers, since it appears to be the simplest cosmology for a hierarchy.     

Oxygen abundances in old disc and halo stars of the Galaxy

Intermediate resolution (/ 25 000) CCD spectra of the oxygen triplet at 7770 Å have been analyzed to determine oxygen abundances in a sample of metal-deficient stars with metallicities covering the range –2.5[Fe/H]–0.2. Important oxygen overabundances ([O/Fe]1) are found in the more metal deficient stars of the sample. We briefly discuss the information that these observations provide about the early nucleosynthesis history and chemical evolution of the Galaxy.