Motion Around The Triangular Equilibrium Points Of The Restricted Three-Body Problem Under Angular Velocity Variation

We study numerically the asymmetric periodic orbits which emanate from the triangular equilibrium points of the restricted three-body problem under the assumption that the angular velocity ω varies and for the Sun–Jupiter mass distribution. The symmetric periodic orbits emanating from the collinear Lagrangian point L ₃, which are related to them, are also examined. The analytic determination of the initial conditions of the long- and short-period Trojan families around the equilibrium points, is given. The corresponding families were examined, for a combination of the mass ratio and the angular velocity (case of equal eigenfrequencies), and also for the critical value ω = 2

Effect of Krein signatures on the stability of relative equilibria

We consider the linear stability of the equilibrium points of the generic rotating potentials U(r), U(r, ), U(r, and U(r, , . The stability analysis is performed using the concept of Krein's signature. This signature is calculated for all eigenvalues of the above potentials. Thereby, the Lagrangian points of the restricted three-body problem and the synchronous satellites of oblate and prolate planets are also studied. We find also the new positions of the eigenvalues for perturbations of the original L ₄ and L ₅ points of Mars, Jupiter, Saturn, Uranus and Neptune. Finally, we study the problem with the mass ratio µ close to the critical value and the stability of geostationary satellites perturbed by the Moon.     

Three-dimensional periodic oscillations generating from plane periodic ones around the collinear Lagrangian points

The three families of three-dimensional periodic oscillations which include the infinitesimal periodic oscillations about the Lagrangian equilibrium pointsL ₁,L ₂ andL ₃ are computed for the value =0.00095 (Sun-Jupiter case) of the mass parameter. From the first two vertically critical (|a _v |=1) members of the familiesa, b andc, six families of periodic orbits in three dimensions are found to bifurcate. These families are presented here together with their stability characteristics. The orbits of the nine families computed are of all types of symmetryA, B andC. Finally, examples of bifurcations between families of three-dimensional periodic solutions of different type of symmetry are given.     

A new examination of absorption line density in the Lα forest of QSOs

The distribution of the absorption lines in the L forest of QSOs is re-examined in relation to both absorption and emission redshifts by analyzing a sample consisting of 20 QSO L forests.On the one hand, the -value in the relationN(z _abs)=N ₀(1+z _abs) is re-estimated by applying the maximum likelihood (ML) method to all the L forests in our sample as a whole as well as to the forests for individual QSOs. The global estimated is 2.20±0.28 and both the quasars with highest and lowestz _em do not individually have a significant effect on it. A non-parametric test on the estimate of from individual QSOs shows that there is no significant difference between the numbers of positive and negative -values. This does not, therefore, favour the idea that for individual QSOsN(z _abs) decreases with increasingz _abs.On the other hand, the possible correlation between the absorption line density and the emission redshift of the QSO itself, which was found by Bianet al. (1986, 1988) and re-discovered independently by Tytler (1987), is shown once again not only by the largerz _em QSOs having larger mean density, but more importantly, at a rather high confidence level, by the fact that, at the same values ofz _abs, QSOs with largerz _em possess a greater number of L absorption lines.A discussion of the inverse effect suggests that it is still hard to say whether this effect really exists.     

Three-dimensional periodic orbits about the triangular equilibrium points of the restricted problem of three bodies

The third-order parametric expansions given by Buck in 1920 for the three-dimensional periodic solutions about the triangular equilibrium points of the restricted Problem are improved by fourthorder terms. The corresponding family of periodic orbits, which are symmetrical w.r.t. the (x, y) plane, is computed numerically for =0.00095. It is found that the family emanating from L₄ terminates at the other triangular point L₅ while it bifurcates with the family of three-dimensional periodic orbits originating at the collinear equilibrium point L₃. This family consists of stable and unstable members. A second family of nonsymmetric three-dimensional periodic orbits is found to bifurcate from the previous one. It is also determined numerically until a collision orbit is encountered with the computations.     

Numerical determination of families of three-dimensional double-symmetric periodic orbits in the restricted three-body problem,II

New families of three-dimensional double-symmetric 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, g ₁, g₂, c 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.     

Singposts of shock waves in the metagalactic medium

Explosions of quasars and young galaxies at large redshifts must cause propagation of blast waves in the metagalactic medium. The shock waves formed can, during the radiative cooling stage, produce dense cold spherical shells around the epicentres of explosions. But, even before that, at the stage of adiabatic expansion, each spherical shock-wave front, if it lies on one line-of-sight with a more distant quasar, can imprint into the quasar spectrum a specific absorption doublet with a distance between the components (in the rest frame) ₀ 3 Å. The L doublet components have a small but the same equivalent widthW ₀ 0.3 Å, the ratioW ₀/₀ weakly depending onW ₀ ifW ₀ is small. We demonstrate here that such absorption L doublets are really present among the L forest in the absorption spectra of distant quasars which are, according to Sargentet al. (1980), of mostly intervening (and not intrinsic) origin. Further accumulation of data on absorption doublets, which can serve as direct indicators of metagalactic shock-waves, may provide valuable information about the physical conditions in the metagalactic gas at large redshifts.     

Resonance and Capture of Jupiter Comets

A number of Jupiter family comets such as Otermaand Gehrels 3make a rapid transition from heliocentric orbits outside the orbit of Jupiter to heliocentric orbits inside the orbit of Jupiter and vice versa. During this transition, the comet can be captured temporarily by Jupiter for one to several orbits around Jupiter. The interior heliocentric orbit is typically close to the 3:2 resonance while the exterior heliocentric orbit is near the 2:3 resonance. An important feature of the dynamics of these comets is that during the transition, the orbit passes close to the libration points L ₁and L ₂, two of the equilibrium points for the restricted three-body problem for the Sun-Jupiter system. Studying the libration point invariant manifold structures for L ₁and L ₂is a starting point for understanding the capture and resonance transition of these comets. For example, the recently discovered heteroclinic connection between pairs of unstable periodic orbits (one around the L ₁and the other around L ₂) implies a complicated dynamics for comets in a certain energy range. Furthermore, the stable and unstable invariant manifold tubes associated to libration point periodic orbits, of which the heteroclinic connections are a part, are phase space conduits transporting material to and from Jupiter and between the interior and exterior of Jupiter's orbit.     

Coronal Leaky Tube Waves and Oscillations Observed with Trace

Leaky tube waves are examined in the context of kink oscillations in coronal loops, observed in recent years using TRACE. It is pointed out that the standard (non-leaky) principal kink mode has a leaky bifurcated counterpart with decay time 4^–4(L/R)² P, where R and L are the loop radius and length, and P is the oscillation period. This is somewhat too long to explain the observed decays, except for very short or thick loops, but may be implicated in the initial excitation. Higher harmonics decay much more rapidly. The external solution takes the form of a wave running nearly parallel to the tube, but with a small outward component. In addition, a number of other leaky modes are described which decay on timescales of seconds, =Ra _e/a ², where a and a _e are the loop and external Alfvén speeds respectively, and which can be identified as being almost radially propagating fast magnetoacoustic waves. These are outside the currently observable range, but are likely to be important energetically.     

Implications of CP violation in hyperon decays

It is generally believed that the only known reaction in whichC _p violation definitely occurs is in the decay of the long-lived neutralK-mesonK _L ^+–,K _L ⁰⁰, andK _L e ^±± v (Christensonet al., 1964: Sivaram, 1982). No attempt has been made to studyC _p violation outside theK-system for quite a long time. Recently,C _p violation effects have been reported in the hyperon decays through the reaction (Bassompierre, 1990) with asymmetry at the level of 10^–3 to 10^–4.In this paper we examine the possible implications of hyperon decays asymmetry in some cosmic-ray sources. We identify cosmic-ray sources where such decays can occur. The signatures for measuring these asymmetries in both the laboratory and cosmic-ray sources are examined.It is found that there is a correlation between these signatures. We conclude that hyperon decays contribute significantly toC _p violation observed in cosmic-ray sources.     

The spatial distribution and birth-rate of pulsars

The distribution of pulsars in the wide range of observed luminosities has been obtained. It is shown that the function of luminosity (FL) within 3×10²⁶L2×10³⁰ erg s^–1 conforms to the power law dN/dL–c ₁ L ^–, where =1.76±0.06. ForL3×10²⁶ erg s^–1, FL changes its inclination and may be approximated as , where ₁ = 0.7±0.2. On the basis of statistical selection, including all pulsars withL>3×10²⁸ erg s^–1, the distribution of pulsars has been investigated as a function of the distance to the centreR and galactic planeZ.The obtained laws of the radial andZ-distribution of pulsars and galactic supernova remnants and also the radial distribution of types I and II supernovae in the models Sb and Sc support the hypothesis of their origin from the objects of the flat subsystem of Population I. Since there are some arguments in favour of a possible connection between supernovae I and the objects of the intermediate component of the Galaxy, one cannot exclude the possibility of supernovae explosions at the end of the evolution of stars with masses of 1.5–2M . It is also shown that pulsars and supernovae are evidently objects that are connected genetically, and, within the limits of statistical error, they have a similar birth-rate.The empirical law of the evolution of a pulsar's luminosity as a function of its true age has been obtained, according to whichL=c ₂ t ^–, wherec ₂=(3.69±3.4)×10³⁵, =1.32±0.11.     

On symmetrical planetary corotations

Exact corotations are equilibrium points in the phase space of the asteroidal elliptic restricted problem of three bodies averaged over the synodic period, at a mean-motions resonance. If the resonant critical angle is =(p+q) _jup –p–q, exact corotations are double resonant motions defined by the conditionsd/dt=0 andd(– _jup )/dt=0. The first condition is characteristic of the periods resonance(p + q) : p and the second one is a secular resonance equivalent to that usually known as thev ₅-resonance. This paper presents the symmetric solutions =0 (mod ), = _jup (mod ). Corotations have a coherence property which is unique in non-collisional Celestial Mechanics: An elementary calculation shows that, in the neighbourhood of these solutions, the motions cluster aroundp independent longitude values and are, in each cluster, as close together as and are close to the equilibrium values.     

The analysis and interpretation of solar X-ray photographs

This paper discusses the methods used to analyse and interpret X-ray filtergrams obtained by solar soft X-ray telescopes such as the S-056 Skylab instrument. First, an appropriate definition of the line-of-sight emission measure _L(T) is developed, and it is shown how the X-ray data may be analysed to obtain an approximation to _L(T). The accuracy of this approximation is severely limited by the mathematical ill-conditioning of the problem, and additional constraints on the solutions must be imposed through the use of a specific model of the coronal region under study. Such a model is also required for the proper interpretation of the results in terms of coronal plasma processes. Examples of such models are provided and the forms of _L(T) derived from them compared with other, semi-empirical forms.The filter ratio method (a simplified form of analysis in which the region under study is assumed isothermal) is discussed. It is shown that in the presence of line-of-sight temperature gradients, the values of effective temperature and emission measure yielded by this method cannot be directly related to the physical state of the plasma and so are of little utility in the study of coronal processes.Now at: Institute for Plasma Research, Stanford University, Via Crespi, Stanford, Calif. 94305, U.S.A.     

The Phase Space Structure Around L4 in the Restricted Three-Body Problem

The phase space structure around L ₄ in the restricted three-body problem is investigated. The connection between the long period family emanating from L ₄ and the very complex structure of the stability region is shown by using the method of Poincarés surface of section. The zero initial velocity stability region around L ₄ is determined by using a method based on the calculation of finite-time Lyapunov characteristic numbers. It is shown that the boundary of the stability region in the configuration space is formed by orbits suffering slow chaotic diffusion.     

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.     

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.     

The Pomeau-Manneville intermittent transition to chaos in hydrodynamic pulsation models

Hydrodynamic simulations of nonlinear pulsation for less-massive cool supergiants have been performed by several authors. Outburst of large amplitude oscillations at times is one of common features of these models. To find out routes of the transition from the limit cycles to the irregular pulsations. We performed hydrodynamic simulation for a series of models of luminosity log(L/L )=3.505, andT _e =5300 K with the range of the mass 1.4M M1.5M . With decreasing the mass, we confirm a transition from limit cycles to irregular oscillations.The nature of the transition is finally specified by examining the dissipation of pulsational kinetic energies in limit cycle models, when pulsations start with larger amplitudes than their limiting pulsations. We find that the rates of dissipation are so small that they might be marginally stable. Furthermore, the oscillation starting with even larger amplitudes gets the kinetic energies until it reaches a limit where the oscillation induces strong shock waves and dissipates its kinetic energy. Thus, we conclude that the model which has the stable limit cycle near the transition has another unstable fixed point above the limit cycle. The transition, therefore, is induced by disappearance of these two fixed points, as the mass, the control parameter in our case, is varied, and is found exactly in aggreement with the intermittency proposed by Pomeau and Manneville as a route to chaos in dissipative systems.     

On Geovenelli's approximate form of j(τ)

The accuracy of the approximate replacement of the frequency meanj() of the total intensityJ () byJ _c+a{J ₀()–J _c} by Geovenelli, whereJ _c is the total intensity at the continuum,J ₀() is that at the line center anda is independent of depth, is studied. Numerical calculations for strong solar lines show very high values of the residual intensities. A slight change in the approximation is then suggested to get a result quite consistent with observational and existing calculated data.     

Redshifts of the brightest X-ray QSO's

The plot of the X-ray luminosity (in 0.5–4.5 KeV band and for Friedmann universe withq ₀=+1) of the brightest X-ray QSO at each redshift against redshift shows that the X-ray luminosity increases more or less monotonically with redshift uptoz3. This result has been attributed to the selection effect known as the volume effect. When this selection effect is taken into account in the optical, radio and X-ray windows of the electromagnetic spectrum, a sample of the brightest X-ray QSO's is obtained which shows a small dispersion in X-ray luminosity: logL ₁=46.15±0.25. The redshift-X-ray flux density plot for this sample gives slopes of both regression lines which agree, at a confidence level of 95% or greater, with the slopes expected theoretically if the redshifts of the QSO's are cosmological in nature.     

Singular points of the polarization tensor

The problem to compute the magnetic field above the chromosphere using data of the vector = B _t/Bt that gives the projected field direction can be solved with different approximations. The field of direction vectors is, however, not the only field accessible to observations. The Stokes parameters, which are components of the radiation tensor, can be measured at each point of the image plane. The directions of the eigenvectors of the radiation tensor define two mutually orthogonal systems of integral curves in the image plane. These families of curves have singular points, which are generally of different type than those of the vector field. When the morphology of H chromospheric fibrils are used to infer the topology of the magnetic field, a similar problem is met, suggesting that singular points should also be present there.