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 .     

Theory of the Trojan asteroids

In the previously published Parts I and II of the paper, the author has constructed a formal long-periodic solution for the case of 11 resonance in the restricted problem of three bodies to 0(m ^3/2), wherem is the small mass parameter of the system. The time-dependencet(, ,m), where is the mean synodic longitude and is related to the Jacobi constant, has been expressed by ahyperelliptic integral. It is shown here that with the approximationm=0 in the integrand, the functiont(, , 0) can be expanded in a series involving standardelliptic functions. Then the problem of inversion can be formally solved, yielding the function (t, , 0).Similarly, the normalized period (,m) of the motion can be approximated by theHagihara hyperelliptic integral (, 0), corresponding tom=0. This integral is also expanded into elliptic functions. Asymptotic forms for (, 0) are derived for 0 and for 1, corresponding to the extreme members of thetadpole branch of the family of orbits.     

The spectra of extended radio sources with a nonuniform magnetic field and hard injection of particles

On the basis of an analytical solution of the diffusion-type kinetic equation for electrons, electron distributions and radiation spectra have been found which result from a hard injection of particles in sources of the core halo type, characterized by spatially nonuniform magnetic fields and diffusion parameters. Such radio sources are shown to possess nonlinear radiation spectra containing universal (=0.5) and diffusion-controlled power-law sections shaped by synchrotron losses, spatial diffusion and radiation conditions of the electrons. The diffusion-controlled sections can be described by spectral indices 0.5<1, if the magnetic field decreases towards the source edge, and by <0.5 where the magnetic field increases.     

The behaviour of the infrared lines of neutral oxygen and the balmer lines of hydrogen in the spectra of early-type stars with emission lines

The equivalent widths of the oxygen lines at 7774 and 8446 and of H (and some H) have been measured for 22 early-type, emission-line stars. A strong correlation between H and 8446 intensities has been found, although there is no such correlation between H and 7774. This confirms the probability that Bowen's mechanism is operative (the neutral oxygen 3³ D state is overpopulated because the excitation energy of Ly- nearly coincides with that of theOi 1025 line). The possibility of using 8446 and H equivalent widths for a comparison of oxygen to hydrogen abundances in these stars is discussed.     

Study of alpha component dynamics in the solar wind using the Prognoz satellite

The analysis of solar wind He⁺⁺ and H⁺ ion distribution functions, collected over five months by the satellite Prognoz 1, shows that these are in general maxwellian but that often tails appear at higher speeds. The existing relation V-T, the observation of ratios of T/Tp 3.83 and V/Vp 1.035 give evidence of preferential He⁺⁺ ion heating and acceleration. The criteria for heating by dissipation of hydromagnetic waves proposed by Barnes and Hung (1973) are tested experimentally. Finally, multifluid models are likely to predict certain observations such as dependence of the velocity ratio V/Vp on the solar wind flux.     

Families of three-dimensional axisymmetric periodic orbits in the restricted three-body problem: Sun-Jupiter case

Families of three-dimensional axisymmetric periodic orbits are determined numerically in the Sun-Jupiter case of the restricted three-body problem. These families bifurcate from the vertical-critical orbits (_v = 1,b _v = 0) of the basic plane familiesi andI. Further the predictor-corrector procedure employed to reveal these families has been described and interesting numerical results have been pointed out. Also, computer plots of the orbits of these families have been shown in conical projections.     

Filter observations of prominences in the D3 and Hα lines

D₃ and H pictures of prominences were obtained with a 21-in. Lyot coronograph and a Fabry-Perot etalon used as a narrow band filter. The monochromatic images of quiescent, quasiquiescent and loop-prominences were studied. The comparison of the isophotes of quiescent and quasi-quiescent prominences in D₃ with those in H shows the similarity of the prominence structure at both wavelength, although there is a strong tendency for an increase in the intensity ratio D₃/H in the upper region of prominences. It seems that it is due to lower temperature in the upper regions of prominences. Probably, the relaxation processes establishing ionization equilibrium play some role. Measurements of the knot intensities of the loop-prominence show strong variations of the intensity ratio D₃/H (more than one order of magnitude).     

Families of three-dimensional plane-symmetric periodic orbits in the restricted three-body problem: Sun-Jupiter case

Two new families of three-dimensional simple-symmetric periodic orbits are determined numerically in the Sun-Jupiter case of the restricted three-body problem. These families emanate from the vertical-critical orbits (_v = 1,c _v = 0)of the familiesi andl of plane symmetric simpleperiodic orbits direct around the Sun and the Sun-Jupiter respectively. Further, the numerical technique employed in the determination of these families has been described and interesting results have been pointed out. Also, computer plots of the orbits of these families have been shown in conical projections.     

The White-Light Limb Flare of 16 August 1989 and its Chromospheric Counterpart

After carefully comparing the white-light (5600±00 Å) and the slit-jaw H images (0.5 Å  passband) of the 2N/X20 white-light flare of 16 August 1989, we found that the H counterpart identification of the bright kernels in continuum by Hiei, Nakagomi, and Takuma (1992) was incorrect. Now we come to the conclusion that none of the two white-light kernels has a corresponding bright H area. Moreover, the loop shapes in white-light are also different from those in H. H loops rose more rapidly than white-light loops. However, their height–time variations on the whole are similar. This indicates that the continuum and chromospheric emissions of the flare presumably come from different plasmas, but may be modulated by some mutual factors, such as large-scale magnetic fields. Analysis of the Hei 10830 Å spectra taken simultaneously with the slit-jaw H images shows that the line-center intensity of Hei 10830 Å doesn't have a good correlation with the intensity of nearby continuum, which supports the above conclusions. In addition, the electron density at the white-light loop top estimated from the continuum around 5600 Å  and 10830 Å  is as high as 10¹²–10¹³ cm^–3.     

A comparison between the helium 10830 Å and the hydrogen Hα chromospheres

Spectroheliograms of resolution about 2 arc sec obtained simultaneously in He 10830 Å and H show in the network a very close agreement in position of dark H mottles and of bright H plage remnants with 10830 Å absorption, though there is not a one-to-one relation between the intensities; the typical intensity in 10830 Å, corrected for overlapping lines, is I 0.91 of the continuum. Some parts of the network do not appear in 10830 Å. This line is much weaker over supergranule centres (I 0.98), though near active regions dark H fibrils coincide with faint 10830 Å fibrils (I0.93–0.98).Operated by the Association of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.     

The structure of periodic solutions in the restricted problem of three bodies

We consider the basic families of plane-symmetric simply-periodic orbits in the Sun-Jupiter case of the plane restricted three-body problem and we study their horizontal and vertical stabilities. We give the critical orbits of these families, corresponding to the vertical stability parameter = 1 and in future communications we shall give the three-dimensional families which emanate from these plane bifurcations.     

Nonsemisimple 1∶1 resonance at an equilibrium

Consider a Hamiltonian system of two degrees of freedom at an equilibrium. Suppose that the linearized vectorfield has eigenvaluesi,i,–i,–i ( , >0) and is not semisimple. In this paper we discuss the real normalization of the Hamiltonian function of such a system. We normalize the Hamiltonian up to 4th order and show how to compute its coefficients. For the planar restricted three body problem atL ₄ the coefficient that plays an important role in the investigation of the qualitative behaviour of periodic solutions near the equilibrium is explicitly calculated.     

Studies of solar flares using optical,X-ray and radio data

I have studied a number of flares for which good X-ray and optical data were available. An average lag of 5.5 s between hard X-ray (HXR) start and H start, and HXR peak and Ha peak was found for 41 flares for which determination was possible. Allowing for time constants the time lag is zero. The peak H lasts until 5–6 keV soft X-ray (SXR) peak. The level of H intensity is determined by the SXR flux.Multiple spikes in HXR appear to correspond to different occurrences in the flare development. Flares with HXR always have a fast H rise. Several flares were observed in the 3835 band; such emission appears when the 5.1–6.6 keV flux exceeds 5 × 10⁴ ph cm^-2 s^-1 at the Earth. Smaller flares produce no 3835 emission; we conclude that coronal back conduction cannot produce the bright chromospheric network of that wavelength.The nearly simultaneous growth of H emission at distant points means an agent travelling faster than 5 × 10³ km s^-1 is responsible, presumably electrons.In all cases near the limb an elevated Ha source is seen with the same time duration as HXR flux; it is concluded that this H source is almost always an elevated cloud which is excited by the fast electrons. A rough calculation is given. Another calculation of H emission from compressed coronal material shows it to be inadequate.In several cases homologous flares occur within hours with the same X-ray properties.Radio models fit, more or less, with field strengths on the order of 100G. A number of flares are discussed in detail.     

Three-Dimensional p–q Resonant Orbits Close to Second Species Solutions

The purpose of this paper is to study, for small values of , the three-dimensional p–q resonant orbits that are close to periodic second species solutions (SSS) of the restricted three-body problem. The work is based on an analytic study of the in- and out-maps. These maps are associated to follow, under the flow of the problem, initial conditions on a sphere of radius around the small primary, and consider the images of those initial points on the same sphere. The out-map is associated to follow the flow forward in time and the in-map backwards. For both mappings we give analytical expressions in powers of the mass parameter. Once these expressions are obtained, we proceed to the study of the matching equations between both, obtaining initial conditions of orbits that will be 'periodic' with an error of the order ^1–, for some (1/3,1/2). Since, as 0, the inner solution and the outer solution will collide with the small primary, these orbits will be close to SSS.     

Material ejecta in a disturbed solar filament

H observations, using the Multichannel Subtractive Double Pass (MSDP) spectrograph operating on the Meudon Solar Tower, have been made of an active region filament which undergoes a disparition brusque. The period of observation was from 10 45 to 13 30 UT on 22 June, 1981. Velocity and intensity fluctuations in H were measured. The proper motions of ejecta were followed allowing their trajectories and vector velocities to be determined. To model the dynamics of ejecta several models using thermal or magnetic driving forces are compared. The most promising model explains the motion as the consequence of magnetic stresses acting on an isolated magnetized plasmoïd in a diverging flux tube.     

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.     

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     

The spectral indices in the optical continuum of the QSOs,BL Lacertae objects and the nuclei of Seyfert,N and normal galaxies

The hypothesis that the optical continua of QSOs, BL Lac objects and the nuclei of Seyfert, N and normal galaxies arise due to incoherent synchrotron radiation from electrons and consequently the flux of radiation in the optical continua of these objects is given by the power law:F(v)v is examined. Following Kinman, spectral indices _B–V and _U–B in(B-V) and(U-B) colours as well as their average and difference have been defined and calculated for samples of 227 QSOs, 32 BL Lac objects and nuclei of 62 Seyfert, 12 N and 7 normal galaxies. Here has been assumed to be an estimate of the spectral index . On the other hand, has been regarded as a measure of departure from the power law. On the basis of this, the distributions of and in QSOs, BL Lac objects and the nuclei of Seyfert, N and normal galaxies have been studied. The value of depends mainly on the balance of the energy loss due to synchrotron radiation and the rate of replenishment of energy by injection of high energy electrons in the radiating region. The increase in the value of and therefore that of indicates that the activity in the object is slowing down and the object is growing older. Assuming that the QSOs, BL Lac objects and the nuclei of Seyfert, N and normal galaxies essentially represent different phases in the evolutionary sequence of extragalactic objects, we suggest that they may be arranged in the sequence: QSOsBL Lac objects Seyferts 1N galaxies Seyferts 2 Normal galaxies in the decreasing order of activity in the core or nuclei of these objects.On leave of absence from the Government Science College, Rajpur, M.P., India.     

Перенос резонансного излучения И случайные ълуждания Фотонов

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Transfer of resonance radiation in infinite medium is considered as a process of random walks of photons. Close relation is shown to exist between the problems of transfer of line radiation and the stable distributions of the probability theory. This relation is used as a basis of a new method for the investigation of the asymptotic properties of the radiation field far from the sources.

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Some statistical properties of Ellerman bombs

One hundred seventy-eight Ellerman bombs were identified and studied with on- and off-band H filtergrams of two active centers, one near disk center and the other near the east limb. The photographs, taken through a 1/4 Å bandpass filter, occasionally attained resolution as fine as 0.3. The mean duration of bombs at H-2 Å was about 13 min near disk center and 11 min near the limb; these times increase slightly when we observe closer to the core of H. Eighty-six percent of the bombs in the near-limb region and 56% in the disk-center region were seen to be accompanied by ejections of dark material; the ejections were 3–30 long. The ejection length appears to be proportional to the bomb size times the bomb lifetime.