The impact of the oblateness of Regulus on the motion of its companion

The fast spinning B-star Regulus has recently been found to be orbited by a fainter companion in a close circular path with orbital period P _b=40.11(2) d. Being its equatorial radius R _e 32% larger than the polar one R _p, Regulus possesses a remarkable quadrupole mass moment Q. We investigate the effects of Q on the orbital period P _b of its companion in order to see if they are measurable, given the present-day level of accuracy in measuring P _b. Conversely, we will look for deviations from the third Kepler law, attributed to the quadrupole mass moment Q of Regulus, to constrain the ratio γ=m/M of the system’s masses. The impact of Q on the orbital period is analytically worked out with a straightforward perturbative approach. The resulting correction P ^Q is compared to other competing dynamical effects. P ^Q and the Keplerian period P ^Kep are expressed in terms of the phenomenologically determined system’s parameters; γ is treated as an unknown. P ^Q is compared to the observational accuracy in measuring the orbital period δ P _b=0.02 d and to the systematic uncertainty δ(P ^Kep) due to the errors in the system’s parameters entering it. The discrepancy ΔP=|P _b?P ^Kep| is examined in order to see for which values of γ it becomes statistically significant. The physical meaning of the obtained range of values for γ is discussed in terms of Q. P ^Q is larger than δ P _b but still smaller than the systematic uncertainty in P ^Kep by two orders of magnitude. The major sources of bias are the velocity semiamplitude K of the motion of the primary and its mass M. Assuming edge-on configuration, i.e. i=90 deg, if γ?0.096 Q would be positive, i.e. Regulus would be prolate, contrary to the observations. If γ?0.078 Q would be negative, but its magnitude would be one-two orders of magnitude larger than the approximate estimate Q≈M(R _p ² ?R _e ² )=?2.4±0.5×10⁴⁹ kg?m². Regulus is the first extrasolar binary system in which the orbital effects of the asphericity of the primary are larger than the observational sensitivity; moreover, no other competing aliasing orbital effects are present. Thus, it is desirable that it will become the object of future intensive observational campaigns in order to reduce the systematic uncertainty due to the system’s parameters below the measurability threshold.     

NLTE formation of the solar spectrum of silicon: Abundance of silicon in a three-dimensional model of the solar atmosphere

We investigated the NLTE formation of the solar spectrum of neutral silicon using 3D hydrodynamic model of the solar atmosphere and realistic atomic model. We show that, within the intergranular region, combined action of the deficit in the source function and excess in opacity due to the overpopulation of the lower Si I levels leads to a considerably higher increase in the central depth D and equivalent width W of these lines as compared to the granules. We have fitted silicon abundances A _W and A _D from the equivalent widths W and central depths D for 65 Si I lines using a 3D model. We show that a total error in the calculated silicon abundance due to neglecting NLTE and 3D effects, as well as the uncertainty in the van der Waals broadening constant γ₆, turns out to be ?0.1 dex. Using a semiclassical theory by Anstee, Barklem, and O’Mara in calculating γ₆ yields a fair coincidence between the values of A _W and A _D , because the average difference A _W — A _D does not exceed 0.01 dex for both NLTE and LTE. When applying the Unsold’s approximation in calculating γ₆ with an enhancement factor E = 1.5, the abundances A _W and A _D proved to be in disagreement with one another. We analyzed the “solar” oscillator strength scale by Gurtovenko and Kostik, as well as the experimental one by Garz and Becker et al. We show that using “solar” oscillator strengths log gfw leads to a minimum trend with the equivalent widths for NLTE abundances A _W , A _D , their difference A _W — A _D , and standard deviations. The NLTE abundance of silicon obtained using solar oscillator strength scale by Gurtovenko and Kostik is A _W ^NTLE = 7.549 ± 0.016. This value is in good agreement with the value of silicon abundance recommended by Grevesse and Sauval for the CI chondrite meteorites.     

The dependence of brightness distribution on the inclination of galaxies

A sample of 92 galaxies with published surface photometric data inB band and inclined randomly to the line of sight is selected and used to compute the photometric parameterQ _N . The dependence ofQ _N on the apparent flatteningq is found and discussed for different morphological groups.Q _N is more sensitive toq in earlier-type galaxies.     

The power-law behaviours of angular spectra of polarized Galactic synchrotron

We study the angular power spectra of polarized Galactic synchrotron in the range 10≲l≤800, at several frequencies between 0.4 and 2.7 GHz and at several Galactic latitudes up to near the North Galactic Pole. Electric- and magnetic-parity polarization spectra are found to have slopes around α_E,B=1.4–1.5 in the Parkes and Effelsberg Galactic-Plane surveys, but strong local fluctuations of α_E,B are found at |b|≃10° from the 1.4 GHz Effelsberg survey. The C_PIl spectrum, which is insensitive to the polarization direction, is somewhat steeper, being α_PI=1.6–1.8 for the same surveys. The low-resolution multifrequency survey of Brouw and Spoelstra [A&AS 26 (1976) 129] shows some flattening of the spectra below 1 GHz, more intense for C_E,Bl than for C_PIl. In no case we find evidence for really steep spectra. The extrapolation to the cosmological window shows that at 90 GHz the detection of E-mode harmonics in the cosmic background radiation should not be disturbed by synchrotron, even around l≃10 for a reionization optical depth τ_ri≳0.05.     

Simulation of f-Mode Propagation Through a Cluster of Small Identical Magnetic Flux Tubes

Motivated by the question of how to distinguish seismically between monolithic and cluster models of sunspots, we have simulated the propagation of an f-mode wave packet through two identical small magnetic flux tubes (R=200 km), embedded in a stratified atmosphere. We want to study the effect of separation d and incidence angle χ on the scattered wave. We have demonstrated that the horizontal compact pair of tubes (d/R=2, χ=0) oscillate as a single tube when the incident wave is propagating, which gives a scattered wave amplitude of about twice that from a single tube. The scattered amplitude decreases with increasing d when d is about λ/2π where λ is the wavelength of the incident wave packet. In this case the individual tubes start to oscillate separately in the manner of near-field scattering. When d is about twice λ/2π, scattering from individual tubes reaches the far-field regime, giving rise to coherent scattering with an amplitude similar to the case of the compact pair of tubes. For perpendicular incidence (χ=π/2), the tubes oscillate simultaneously with the incident wave packet. Moreover, simulations show that a compact cluster oscillates almost as a single individual small tube and acts more like a scattering object, while a loose cluster shows multiple-scattering in the near field and the absorption is largest when d within the cluster is about λ/2π. This is the first step to understand the seismic response of a bundle of magnetic flux tubes in the context of sunspot and plage helioseismology.     

Distribution of gaps in emission line redshifts of QSOs

A gap in the distribution of a parameter is simply the absence of the parameter for the values corresponding to the gap. The gap in the emission line redshift (z) of QSOs thus represents absence of QSOs with emission line redshift values corresponding to the gap region. Gaps in emission line redshifts of QSOs have been analysed statistically with updated data consisting of 1549 values. The study indicates: (i) There is a critical redshiftz _c =2.4, which separates two distinct phases in the creation of QSOs. Forz>z _c , the creation appears to have been a slow process. Atz?z _c there was a triggering action which produced a burst of QSOs simultaneously. Forz _c , the rate of production of QSOs have been fast. (ii) The distribution of gaps atz _c ; appear to be consequence of periodicities, provided the periodicities involved are perfect and the redshift values are accurate. (iii) The distribution of gaps atz>z _c are not random, but follow a definite trend.     

The circular polarization of sources of synchrotron radiation

The degree of circular polarizationp _c is calculated for two models of a source of synchrotron radiation:

1. A source with an inhomogeneous magnetic field and isotropic angular distribution of the electrons with respect to the magnetic field;
2. A source with a homogeneous magnetic field and anisotropic angular distribution of the electrons in which the anisotropy of angular distribution substantially increases with the electron energy.

The first model can be used to describe extended radio-sources; and the second, to describe compact radio-sources. For those sources, whose observed polarization properties correspond to the first model, we obtain an integral equation which connects the observed distribution of the sources with the extent of their linear and circular polarization (p _l andp _c ) and the unknown distribution of the sources over the strengthB and the degree of homogeneity ?=(B ₀/B)² of the magnetic field;B ₀ is a homogenous field,B ₀?B. A solution of the integral equation obtained is found for a particular case. This solution makes it possible to determine the distribution of different types of sources over ? if the distribution of these sources in the extent of linear polarization is known. The formulae obtained make it possible to indicate which sources with a known degree of linear polarization should be expected to exhibit highest circular polarization. In the discussion of the first model the question is raised as to the information one can get about the magnetic field by using observations of both linear and circular polarization for a separate source, and for a number of sources. It is shown that the determination of the most probable values ofB and ? in a separate source based on the known values ofp _l andp _c for the source, is possible only if one knows the distribution overB and ? of the sources of the type to which the source in question belongs. The observational data now available make it possible to find the distribution of the sources only over ?. Since the distribution overB and ? is at present unknown, even a very strong upper limit forp _c in the case of a separate source does not enable us to give an exact upper limit for the strength of the magnetic field in this source. In the first model the upper limit for the magnetic field can be obtained only if the upper limit ofp _c is known for a certain number of sourcesN, withN?1. This limit allows for much stronger fields than are usually admitted. This last fact should be taken into consideration when one deals with the results of observations of circular polarization in sources with strong magnetic fields. The first model presents some difficulties when we compare it with observations of some compact sources. The second model can explain why one observes in these sources a violation of the lawp _c ～v ^?1/2 and a change of sign inp _c when the frequency of the observationsv changes.     

Reconstruction of f(R,T) gravity in anisotropic cosmological models of accelerating universe

In this paper, we search the existence of Bianchi type I cosmological model in f(R,T) gravity, where the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar R and of the trace of the stress-energy tensor T. We obtain the gravitational field equations in the metric formalism, and reconstruct the corresponding f(R,T) functions. Attention is attached to the special case, f(R,T)=f ₁(R)+f ₂(T) and two examples are assumed for this model. In the first example, we consider the unification of matter dominated and accelerated phases with f(R) gravity in anisotropic universe, and in the second instance, model of f(R,T) gravity with transition of matter dominated phase to the acceleration phase is obtained. In both cases, f(R,T) is proportional to a power of R with exponents depending on the input parameters.     

The formation of ring galaxies

The conditions under which a head-on collision between a disk galaxy and a spherical galaxy can lead to ring formation are investigated, using the impulsive approximation. The spherical galaxy is modeled as a polytrope of indexn=4 and radiusR _S and the disk galaxy as an exponential disk whose surface density is given by \(\sigma (r) = \sigma _c e^{ - 4r/R_D } \) , where σ _c is the central density andR _D is the radius of the disk. The formation and properties of the rings are closely related to the fractional change in binding energy of the disk galaxy, given by ΔU/?U?=γ _D β _D , where (GM _S ² R _D )/(V ² M _D R _S ² ),M _S andM _D being the masses of the spherical and disk galaxies, respectively, and β _D ≡β _D (n, σ, ?,i) is a function of the models of the two galaxies, the ratio of the radii of the two galaxies ?=R _S /R _D , and the angle of inclinationi, of the disk to the direction of relative motion of the two galaxies. Calculations are made for the caseR _S =R _D . Since practically the entire mass of the spherical galaxy, for the chosen model, lies within 1/3 of its radius, the radius of the spherical galaxy is effectively \(\tfrac{1}{3}\) that of the disk galaxy. It is found that as a result of the collision, the innermost and the outer parts of the disk galaxy are not much affected, but the intermediate region expands and gets evacuated, leading to the crowding of stars in a preferential region forming a ring structure. The rings are best formed for a normal, on-axis collision. For this case, rings form when ΔU/|U| lies between \(\tfrac{1}{2}\) and 2, while they are very sharp and bright when ΔU/|U| lies between \(\tfrac{1}{2}\) and 1. Within this range, as ΔU/|U| increases, the rings become sharper and their positions shift outwards with respect to the centre of the disk galaxy. The relationship $$\gamma _D = 0.0016 + 0.045s_{{\text{max}}}^2 ,$$ wheres _max is the radial distance of the density maximum of the ring from the centre of the disk galaxy (measured in terms of the radius of the disk galaxy as unit) enables us to finds _max from γ _D and vice versa, and interpret some prominent ring galaxies. The effect of introducing a bulge to the disk is to distribute the tidal disruptive effects more evenly and, hence, reduce the sharpness of the ring.     

Photoionization features of the ground and excited levels of Cl II and benchmarking with experiment

This report presents i) various characteristic features in photoionization cross sections (σ_PI) of Cl II + hν  →  Cl III + e for many fine structure levels of Cl II, 392 in total with n ≤ 10 and l ≤ 9, ii) comparison of features with those observed in an experiment carried out at the Advanced Light Source at Lawrence Berkeley National Lab, and iii) partial photoionization cross sections of the ground level for ionization leaving the core ion in to various excited levels for applications in plasma modeling. The features correspond to resonant structures, the shape of the background, and their interference effect in σ_PI of this near neutral ion Cl II with 16 electrons. σ_PI for the 5 levels of the ground configuration 3s²3p⁴(³P_0,1,2, ¹D₂, ¹S₀) of Cl II show regions of narrow Rydberg resonances at and near threshold energies, and resonant structures at higher energies in contrast to typical smooth decrease in the background. Various other features in σ_PI of levels of excited equivalent electron states and broad Seaton resonances in single valence electron excited levels are illustrated with examples. Comparison of calculated σ_PI of the 15 lowest levels with the combined features of the measured photoionization spectrum shows excellent agreement by reproducing and thus identifying them to the levels that they belong to. The calculations were carried out in relativistic Breit-Pauli R-matrix (BPRM) method using a close coupling wave function expansion of 45 levels up to 4s of the core ion Cl III. These levels were optimized using a set of 12 configurations going up to orbital 5s, 3s²3p³, 3s3p⁴, 3p⁵, 3s²3p²3d, 3s²3p²4s, 3s²3p²4p, 3s²3p²4d, 3s²3p²4f, 3s²3p²5s, 3s3p³3d, 3s²3p3d², 3p⁴3d producing 283 levels of Cl III. The autoionizing resonances are delineated with a fine energy mesh to observe the fine structure effects. The present results will provide high precision parameters for various applications involving this less studied ion.     

Near-opposition limb darkening of solids of planetary interest

This paper presents a laboratory study of the limb darkening near opposition, of particulate materials of planetary interest and concentrates on the wavelength dependence of this limb darkening. We find that near zero phase the scattering properties of most particulate materials can be described adequately by Minnaert's law. However, there are materials for which such a representation is totally inadequate. Examples are bronzite and graphite, materials that tend to fracture into flakes having mirrorlike surfaces. In addition, there are materials, such as olivine, whose scattering properties within deep absorption bands show definite departures from Minnaert's law at large angles of incidence or emission. Our Minnaert parameters, k and B₀, were measured at a phase angle of α = 4°. For samples of comparable surface texture and roughness, k and B₀ are approximately linearly related, k increasing as B₀ increases. Very dark materials tend to have k ～ 0.5 to 0.6, while very bright materials tend to have k ～ 1. The linear relationship between k and B₀ can be explained in terms of the varying importance of multiple scattering in the surface layer. Thus for materials for which B₀ is strongly wavelength dependent, so is k. For example, for olivine, k varies from 0.73 to 0.87 between 0.4 and 1.2 μm. These variations are closely correlated with those in B₀: the value of k is relatively high outside of absorption bands and relatively low within them. For bright materials, k is very sensitive to surface roughness. For example, for BaSO₄ powder, k can be changed from ～1.0 to ～0.8 by this effect alone, a fact which has relevance to the photometry of frost-covered satellites. For dark materials, the effects of surface roughness on k are smaller and more subtle.     

Instability of an area-preserving polynomial mapping

Fixed points and eigencurves have been studied for the Hénon-Heiles mapping:x′=x+a (y?y ³),y′=y(x′?x′ ³). Eigencurves of order 21 proceed rapidly to infinity fora=1.78, but as ‘a’ decreases, they spiral around the origin repeatedly before escaping to infinity. Fixed pointsx _f on thex-axis have been located for the range 1≤a≤2.4, for ordersn up to 100. Their locations vary continuously witha, as do the eigencurves, and hyperbolic points remain hyperbolic. Forn=3 and 2.4≥a≥2.37, a very detailed study has been made of how escape occurs, with segments of an eigencurve mapping to infinity through various escape channels. Further calculations with ‘a’ decreasing to 2.275 show that this instability is preserved and that the eigencurve will spiral many times around the origin before reaching an escape channel, there being more than 34 turns fora=2.28. The rapid increase of this number is associated with the rapid decrease of the intersection angle between forward and backward eigencurves (at the middle homoclinic point), with decreasing ‘a’, this angle governing the outward motion. By a semi-topological argument, it is shown that escape must occur if the above intersection angle is nonzero. In the absence of a theoretical expression for this angle, one is forced to rely on the numerical evidence. If the angle should attain zero for a valuea=a _c>a_m,wherea _m .is the minimum value for which the fixed points exist, then no escape would be possible fora _c However, on the basis of calculations by Jenkins and Bartlett (1972) forn=6, and the results of the present article forn=3, it appears highly probable thata _c=a_m,and that escape from the neighborhood of a hyperbolic point is always possible. If there is escape from the hyperbolic fixed point forn=4,a=1.6, located atx _f=0.268, then the eigencurve must cross the apparently closed invariant curve of Hénon-Heiles which intersects thex-axis atx?±0.4, so that this curve cannot in fact be closed.     

A Statistical Analysis on the Color and Orbit Parameters of Centaurs

In order to study the statistical characters and physical significance of the color indexes and orbital parameters of Centaurs, we have analyzed the color and orbit parameter correlations by using statistical methods according to the known color indexes of 39 Centaurs and their respective orbital parameters. We find that all the correlations of B − V vs. V − R, V − R vs. R − I, B − V vs. R − I and B − R vs. R − I are strong. The Centaurs exhibit obvious redblue double-color property, and the point of demarcation is close to B − R = 1.4. The distributions of the various color indexes of Centaurs exhibit the normal distribution. For the correlations between colors and orbital parameters, except for the weak correlations between B − V and the orbital inclination i as well as between V − R and the orbital semi-major axis a, no evident correlation has been found between color indexes and orbital parameters.     

Characteristics of activity energetics of the UV cet-type flare stars

A quantitative comparison of energetics of different manifestations of the activity of the UV Cet-type flare stars — sporadic flares, quiet chromospheres and coronae and stellar spots — have been carried out. On the basis of a statistical analysis of about 1800 flares registered for 23 flare stars, the energy spectra of flares have been constructed and time-averaged powers of optical radiation of flares —L _flares — have been estimated. By use of spectroscopic observations of 26 flare stars between flares, the total radiation in Balmer lines of quiet chromospheres —L _Bal — have been found. On the basis of these data and the soft X-ray observations of 29 flare stars, the mean of the ratiosL _flares/L _bol,L _Bal/L _bol andL _X /L _bol have been found to be equal to (1–2)×10^?4. By use of photometric observations for 9 stars, the total starspot radiation deficits —L _spots — have been estimated as (4×10^?3?6×10^?2)×L _bol. Discussions of probable underestimates in the valuesL _flares,L _Bal,L _X , andL _spots leads to the conclusion that the total power of non-equilibrium processes that take place in flare star atmospheres is close to the total starspot radiation deficit in these stars.     

Numerical Investigation of the Galactic Problem by Computing Families of Periodic Solutions

The galactic dynamical system expressed by a third-order axisymmetric polynomial potential is investigated numerically by computing periodic solutions. We define as Sthe compact set of initial conditions generating bounded motions, and as S ^p , with S ^p ? S, the countable set of all initial conditions generating periodic solutions. Then, we consider the subsets S _s ^p and S _a ^p of S ^p , where S _s ^p ∪ S _a ^p = S ^p , S _s ^p S _a ^p = Ø, the first of which corresponds to symmetric periodic solutions, and the second to asymmetric solutions. Then, we approximate the set S _s ^p , leaving treatment of the set S _a ^p of asymmetric solutions for a future publication. The set S _s ^p is known to be dense in S (‘Last Geometric Theorem of Poincar;’, Birkhoff, 1913). Using a computer programme capable to locate all elements of the set S _s ^p that generate symmetric periodic solutions that re-enter after intersecting the axis of symmetry from 1 to ntimes. The results of the approximation of S _s ^p in the total domain and in the sample sub-domains of zooming, we present in graphical form as family curves in the (x, C) plane. The solutions located with the largest periods re-enter after 440 galaxy revolutions while the families calculated fully (initial conditions, period, energy, stability co-efficient) include solutions that re-enter after 340 galaxy revolutions. To advance further the approximation of the set S _s ^p thus obtained, we applied the same procedure inside eight sub-domains of the domain Sinto which we ‘zoomed’ through selection of finer search steps and double maximum periods. The family curves thus calculated presented in the (x, C) plane do not intersect anywhere in some sub-domains and their pattern resembles that of laminar flow. In other sub-domains, however, we found family curves from which branching families emanate. The concepts of completeand non-completeapproximation of S _s ^p in sub-domains of laminar and sub-domains with branching family curves, respectively, is introduced. Also, the concept of basic family of order1, 2, ..., n, are defined. The morphology of individual periodic solutions of all families is investigated, and the types of envelopes found are described. The approximate set S _s ^p was also checked by computing Poincar; sections for energy values corresponding to the mean energy range of the eight sub-domains of zooming mentioned above. These sections show that most parts of the compact domain in Sgenerating non-periodic but bounded solutions correspond to with well-shaped tori that intersect the x-axis, a fact that implies that dominant to exclusive type of periodic solutions are the symmetric ones with two normal crossings of this axis. The presence of non-symmetric periodic solutions as well as of chaotic regions is encountered. All calculations reported here were performed using the variable step R-K 8th-order direct integration and setting the allowable energy variation Δ C= |C _start? C _end| < 10^?13. The output, consisting of many thousands of families and their properties (initial conditions, morphology, stability, etc.), is stored in a directory entitled ‘Atlas of the Symmetric Periodic Solution of the Galactic Motion Problem’.     

Shadow mechanism and the opposition effect of brightness of atmosphereless celestial bodies

We consider the Irvine-Yanovistkii modification of the shadow model developed by Hapke for the opposition effect of brightness. The relation between the single scattering albedo ω and the transparency coefficient of particles κ is suggested to be used in the form κ = (1 ? ω) ⁿ , which allows the number of unknowns in the model to be reduced to two parameters (the packing density of particles g and ω) and the single-scattering phase function χ(α). The analysis of spectrophotometric measurements of the moon and Mars showed that the data on the observed opposition effect and the changes in the color index with the phase angle α well agree if the values of n = 0.25 and g = 0.4 (the moon) and 0.6 (Mars) are assumed in calculations. When being applied to asteroids of several types, this method also yielded a satisfactory agreement. For the E-type asteroids, the sets of parameters are [g = 0.6, ω = 0.6, A _g = 0.21, and q = 0.83] or [g = 0.3, ω = 0.4, A _g = 0.15, and q = 0.71] under the Martian single-scattering phase function; for the M-type asteroids, it is [g = 0.4, ω ≤ 0.1, A _g ≤ 0.075, and q ≤ 0.42] under the lunar single-scattering phase function; for the S-type asteroids, it is [g = 0.4, ω = 0.4, A _g = 0.28, and q = 0.49] under the lunar single-scattering phase function; and for the C-type asteroids, it is [g = 0.6, ω ≤ 0.1, A _g ≤ 0.075, and q = 0.43] under the modified lunar single-scattering phase function. The polarization measurements fulfilled by Gehrels et al. (1964) for the bright feature on the lunar surface, Copernicus (L = -20°08′, φ = +10°11′), at a phase angle α = 1.6° revealed the deviations in the position of the polarization plane from that typical for the negative branch. They were 22° and 12° in the G and I filters, respectively. At the same time, the deviation was within the error (±3°) in the U filter and for the dark feature Plato (L = -10°32′, φ = +51°25′), which can be caused by the coherent mechanism of the formation of the polarization peak.     

Instabilities and bifurcations of the families of collision periodic orbits in the restricted three-body problem

By using Birkhoff's regularizing transformation, we study the evolution of some of the infinite j-k type families of collision periodic orbits with respect to the mass ratio μ as well as their stability and dynamical structure, in the planar restricted three-body problem. The μ-C characteristic curves of these families extend to the left of the μ-C diagram, to smaller values of μ and most of them go downwards, although some of them end by spiralling around the constant point S* (μ=0.47549, C=3) of the Bozis diagram (1970). Thus we know now the continuation of the families which go through collision periodic orbits of the Sun-Jupiter and Earth-Moon systems. We found new μ-C and x-C characteristic curves. Along each μ-C characteristic curve changes of stability to instability and vice versa and successive very small stable and very large unstable segments appear. Thus we found different types of bifurcations of families of collision periodic orbits. We found cases of infinite period doubling Feigenbaum bifurcations as well as bifurcations of new families of symmetric and non-symmetric collision periodic orbits of the same period. In general, all the families of collision periodic orbits are strongly unstable. Also, we found new x-C characteristic curves of j-type classes of symmetric periodic orbits generated from collision periodic orbits, for some given values of μ. As C varies along the μ-C or the x-C spiral characteristics, which approach their focal-terminating-point, infinite loops, one inside the other, surrounding the triangular points L₄ and L₅ are formed in their orbits. So, each terminating point corresponds to a collision asymptotic symmetric periodic orbit for the case of the μ-C curve or a non-collision asymptotic symmetric periodic orbit for the case of the x-C curve, that spiral into the points L₄ and L₅, with infinite period. All these are changes in the topology of the phase space and so in the dynamical properties of the restricted three-body problem.     

The influence of a turbulent region on the flux of auroral electrons

The influence of low-frequency electrostatic turbulence on the flux of precipitating magnetospheric electrons is analyzed in the framework of the quasilinear kinetic equation. It is shown that an electron population in a turbulent region, with an electric field parallel to the ambient magnetic field, can be separated into two parts by introducing a pitch angle dependent runaway velocity v_r(θ). Lower energy electrons with parallel velocity v_∥ < v_r are effectively scattered by plasma waves, so that they remain in the main population and are subjected to an anomalous transport equation. A distribution function f ∞ v^?4 (or the particle flux vs energy J ∞ E^?1) is established in this velocity range. Faster electrons with v_∥ ? v_r are freely accelerated by a parallel electric field, so that they contribute directly to hot electron fluxes which are observed at ionospheric altitudes. New expressions are derived for the magnetic-field aligned current and the electron energy flux implied by this model. These expressions agree well with empirical relations observed in auroral inverted-V structures.     

Dynamical portrait of the Lixiaohua asteroid family

In this paper we present a comprehensive analysis of the dynamics in the region of the (3556) Lixiaohua asteroid family. The family lies in a particularly interesting region of the phase space, crossed by several two-body and three-body mean motion resonances. Also, members of this family can have close encounters with large asteroids, such as Ceres. We have identified the mean motion resonances which contribute to the long-term dynamical evolution of the family and our results confirm that the members of this family can be classified into a number of groups, exhibiting different dynamical behavior. We show for the first time that in the Lixiaohua region, apart from the chaotic diffusion in proper eccentricity and inclination (e _p and I _p ), there is at least one extended chaotic zone where several resonances overlap, thus giving rise to chaotic diffusion in proper semi-major axis (a _p ) as well. Using a code of Monte Carlo type, we simulate the evolution of the family, according to the model which combines the chaotic diffusion (in a _p , e _p and I _p ), Yarkovsky/YORP thermal effect and random walk in a _p due to the close encounters with massive asteroids. These simulations show that all these effects should be taken into account in order to accurately explain the observed distribution of family members in the space of proper elements, although a “minimal” model that accounts for chaotic diffusion in (e _p , I _p ), Yarkovsky-induced drift in a _p and random walk in a _p due to the close encounters with the most massive asteroids is enough to grossly characterize the shape of the family.     

Investigation of solitary waves in warm plasma for smaller order relativistic effects with variable pressures and inertia of electrons

In the two component relativistic plasmas subject to pressure variation of adiabatic electrons and isothermal ions, both compressive and rarefactive KdV solitons are established in a quite different physical plasma model. It is desirable to define c _s in a new way to substantiate the validity of the model under relativistic effects. The corresponding mathematical condition is also determined, which is a new report of this kind. It is also interesting to report that the relativistic rarefactive solitons cease to exist below some critical ion initial streaming speed v _i0 for a fixed temperature α and electron streaming speed v _e0. Besides, higher initial flux v _i0 of ions under constant temperature is observed to generate higher speed v _i at the passage of time which causes to increase (in relativistic sense) its mass diminishing thereby the growth of soliton amplitudes.