A new light on the L 1, L 2 Lagrangian points

The relation between the locations of L ₁, L ₂ Lagrangian points and the boundary to their respective satellite system is brought forth, in that, the Lagrangian points L ₁, L ₂ are seen to lie just on the boundary to their respective satellite system.     

Bifurcation points and intersections of families of periodic orbits in the three-dimensional restricted three-body problem

Intersections of families of three-dimensional periodic orbits which define bifurcation points are studied. The existence conditions for bifurcation points are discussed and an algorithm for the numerical continuation of such points is developed. Two sequences of bifurcation points are given concerning the family of periodic orbits which starts and terminates at the triangular equilibrium pointsL ₄,L ₅. On these sequences two trifurcation points are identified forµ = 0.124214 andµ = 0.399335. The caseµ = 0.5 is studied in particular and it is found that the space families originating at the equilibrium pointsL ₂,L ₃,L ₄,L ₅ terminate on the same planar orbitm _1v of the familym.     

The elliptical three-body problem. Triangular solution

The restricted three-body problem with eccentric orbit is reviewed and the positions of the triangular Lagrangian points (L₄, L₅) are determined. It is put in evidence the fact the fact L₄ and L₅ are situated at the corners of an isoscales triangle: AB = BC = 1 − e²/)1 + e cos ν )4/3 and AC = 1 − e²/)1 + e cos ν )     

Periodic motion around stable collinear equilibrium point in the photogravitational restricted problem of three bodies

In a binary system with both bodies being luminous, the inner collinear equilibrium pointL ₁ becomes stable for values of the mass ratio and radiation pressure parameters in a certain region. The kind of periodic motions aroundL ₁ is examined in this case. Second-order parametric expansions are given and the families of periodic orbits generated fromL ₁ are numerically determined for several sets of values of the parameters. Short- and long-period solutions are identified showing a similarity in the character of periodicity with that aroundL ₄. It is also found that the finite periodic solutions in the vicinity ofL ₁ are stable.     

The equilibrium solutions of the restricted problem of three rigid bodies with variable mass

In this paper we consider the circular planar restricted problem of three rigid bodiesS _i(i=1, 2, 3), two of them are axisymmetric ellipsoids and a third bodyS ₃ is a spherical satellite with decreasing mass, under the gravitational forces. The effect of small perturbations in the Coriolis force and the centrifugal forces on the location of equilibrium points has been studied. It is found only in the case when the primaries have equal differences between their respective principal moments of inertial the pointsL ₄ andL ₅ form nearly equilateral tringles with the primaries. The equilibrium pointsL ₁,L ₂,L ₃ remain collinear an ies on the line joining the primaries.     

Broad-Band Radio to X-Ray Properties of Seyfert Galaxies

We report on the statistical analysis of a sample of AGNs (Seyferts and LINERs) selected from an RBSC-NVSS sample of the brightest X-ray sources. The sample is large enough and unbiased for a detailed study of multi-frequency (radio to X-ray) properties of AGN. There are no significant differences in X-ray, optical, far-infrared, and radio powers, core dominance, radio loudness, radio spectral index, and two-point radio-to-optical and optical-to-X-ray spectral indices of Seyfert types 1 and 2. These findings strongly support the unification model, and the observed differences between Sy1 and Sy2 are due to both the variable obscuration and different geometric orientation effects. The correlation between the X-ray and radio emission over many decades is primary and indicates the intrinsically similar origin of both radiations in Seyfert galaxies, powered mainly by AGN, rather than compact starbursts. For Seyfert galaxies we found that the slopes of the relations L _X - L _R and L _X - L _B are flat, which indicate the existence of components unrelated with X-ray. Perhaps only for LINERs is the observed L _X - L _R relation completely due to AGNs. The Sy1, Sy2, and LINER galaxies show different slopes in the relation L _X - L _1.4, and the relative contribution of extended radio components may be the cause of these differences. The possible effects of unresolved extended radio and X-ray components are quantitatively discussed.     

Numerical simulation of the electron capture process in a magnetar interior

In a superhigh magnetic field, direct Urca reactions can proceed for an arbitrary proton concentration. Since only the electrons with high energy E (E>Q, Q is the threshold energy of inverse β-decay) at large Landau levels can be captured, we introduce the Landau level effect coefficient q and the effective electron capture rate Γ _eff . By using Γ _eff , the values of L _X and L _ν are calculated, where L _X and L _ν are the average neutrino luminosity of AXPs and the average X-ray luminosity of AXPs L _X , respectively. The complete process of electron capture inside a magnetar is simulated numerically.     

Radio loudness of high-redshift (z≥3) quasars from the Sloan Digital Sky Survey

Using a homogenous sample of 1962 quasars with redshift 3.0≤z≲4.5 drawn from the Sloan Digital Sky Survey (SDSS), we study the relationships between radio loudness, virial black hole (BH) mass and Eddington ratio (accretion rate relative to the Eddington limit). For the radio-detected objects, we find a significant (>99.5 per cent) anticorrelation between radio loudness R parameter and BH mass, consistent with previous studies of low-redshift radio-loud quasars. The truly radio-loud quasars (R>30) are found to be confined to M _BH≲10¹⁰ M_⊙ within our sample. We also find that R is only weakly correlated with Eddington ratio L _bol/L _Edd. Combined with previous results on the low-redshift R−L _bol/L _Edd relation, this result indicates no strong L _bol/L _Edd dependence of R at L _bol/L _Edd≳10⁻². On the other hand, the large scatter in these relationships suggests that other physical properties such as BH spin and quasar clustering must also play an important role in quasar radio emission.     

Structure of X-ray irradiated stellar wind

The effect of X-ray irradiation on the line-driven stellar wind in X-ray binary systems is studied. The product of the X-ray luminosityL _x and the densityn in the wind is a measure to distinguish between the optically thin and thick wind regimes. For an X-ray source of the radiation temperature of 10 keV, the critical value(L _x n) _c , is of the order of 10³⁷ erg s^–1 cm^–3; hence, most of wind-fed X-ray sources lie in the optically thick wind regime because ofL _x n>(L _x n) _c . Then the wind structure is determined not only by the parameter =(L _x /nR ²),R the distance from the X-ray source, but also by the optical thickness due to helium. The formation of fully ionized helium region depends onL _x andn in a way different from that of the Strömgren sphere. In such an optically thick wind, the region behind the He II ionization boundary is little affected by X-ray irradiation and the trace elements remain to be responsible for wind acceleration. Thus, its location is important for the structure of the wind and the interpretation of various phenomena in objects such as wind-fed X-ray pulsars.     

Effects of partial frequency redistribution functions RII,RIII and RV on source functions

The effects of partial frequency redistribution on the formation of spectral lines have been studied. We considered the angle-averaged R_II, R_III andR _v types of redistribution with isotropic phase function, Transfer equation with plane-parallel geometry is solved in isothermal atmospheres. For an atmosphere with constant thermal sources, the frequency-dependent source function S_L (R _v) lies below S_L (R_III) but above S_L(R _III) in the line wings.     

Variation of stellar coronal activity with age

We investigate the variation in coronal activity of late-type stars with age. We determine the age of the star from the colour-colour diagram (U-B vsB-V). We show that the ratio of X-ray luminosity (L _x) to bolometric luminosity (L _bol) is well correlated with age over several orders of magnitude variations. We find thatL _x/L _bol) decreases with age with a power of –1.55. This is consistant with the expected results from the studies of the variation of the corona with chromosphere and the dependence of chromosophere on age.     

X-ray and radio emission by clusters of galaxies

analysis of new X-ray and optical data confirms that the X-ray luminosityL _x of a cluster of galaxies is strongly dependent on its richness. The radio powerP ₁₄₄₅ of clusters at 1445 MHz is independent of richness, but is greater on the average for clusters with dominant cD galaxies than for those withou. The radio emission depends on the activity of one of the brightest galaxies; dominant cD galaxies are often responsible for radio emission, especially if they have double or multiple nuclei. For a given richness, high values ofL _x andP ₁₄₄₅ sometimes occur together, but no strict correlation betweenL _x andP ₁₄₄₅ exists. Possible tests of thermal-bremsstrahlung and inverse-Compton theories for the X-ray emission are mentioned.     

Large kinetic power in FRII radio jets

We investigate the total kinetic powers (L _j) and ages (t _age) of powerful jets of four FR II radio sources (Cygnus A, 3C 223, 3C 284, and 3C 219) by the detail comparison of the dynamical model of expanding cocoons with observed ones. It is found that these sources have quite large kinetic powers with the ratio of L _j to the Eddington luminosity (L _Edd) resides in 0.02<L _j/L _Edd<10. Reflecting the large kinetic powers, we also find that the total energy stored in the cocoon (E _c) exceed the energy derived from the minimum energy condition (E _min): 2<E _c/E _min<160. This implies that a large amount of kinetic power is carried by invisible components such as thermal leptons (electron and positron) and/or protons.     

Asymptotic Orbits at the Triangular Equilibria in the Photogravitational Restricted Three-Body Problem

We study numerically the asymptotic homoclinic and heteroclinic orbits associated with the triangular equilibrium points L ₄ and L ₅, in the gravitational and the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated to these critical points, are also presented. Hundreds of asymptotic orbits for equal mass of the primaries and for various values of the radiation pressure are computed and the most interesting of them are illustrated. In the Copenhagen case, which the problem is symmetric with respect to the x- and y-axis, we found and present non-symmetric heteroclinic asymptotic orbits. So pairs of heteroclinic connections (from L ₄ to L ₅ and vice versa) form non-symmetric heteroclinic cycles. The termination orbits (a combination of two asymptotic orbits) of all the simple families of symmetric periodic orbits, in the Copenhagen case, are illustrated.     

Trajectories of <Emphasis Type="Italic">L</Emphasis><Subscript>4</Subscript> and Lyapunov Characteristic Exponents in the Generalized Photogravitational Chermnykh-Like problem

This paper describes design of the trajectory and analysis of the stability of collinear point L ₂ in the Sun-Earth system. The modified restricted three body problem with additional gravitational potential from the belt is used as the model for the Sun-Earth system. The effect of radiation pressure of the Sun and oblate shape of the Earth are considered. The point L ₂ is asymptotically stable up to a specific value of time t correspond to each set of values of parameters and initial conditions. The results obtained from this study would be applicable to locate a satellite, a telescope or a space station around the point L ₂.     

Error estimates with L1 solutions

A method is given, based on the pseudoinverse of the equations of condition, to obtain error estimates for the solution in the normL ₁ of an over-determined linear system. The computational labor to obtain the errors, while not trivial, is less than that for various competing methods, particularly if there are many more equations of condition than unknowns. The error estimates for anL ₁ solution are substantially larger than those for a least squares solution of the some data. It is suggested that a complete discussion of a linear system include at least bothL ₁ and least squares solutions with their respective errors and the condition number of the linear system.     

Artificial halo orbits for low-thrust propulsion spacecraft

We consider periodic halo orbits about artificial equilibrium points (AEP) near to the Lagrange points L ₁ and L ₂ in the circular restricted three body problem, where the third body is a low-thrust propulsion spacecraft in the Sun–Earth system. Although such halo orbits about artificial equilibrium points can be generated using a solar sail, there are points inside L ₁ and beyond L ₂ where a solar sail cannot be placed, so low-thrust, such as solar electric propulsion, is the only option to generate artificial halo orbits around points inaccessible to a solar sail. Analytical and numerical halo orbits for such low-thrust propulsion systems are obtained by using the Lindstedt Poincaré and differential corrector method respectively. Both the period and minimum amplitude of halo orbits about artificial equilibrium points inside L ₁ decreases with an increase in low-thrust acceleration. The halo orbits about artificial equilibrium points beyond L ₂ in contrast show an increase in period with an increase in low-thrust acceleration. However, the minimum amplitude first increases and then decreases after the thrust acceleration exceeds 0.415 mm/s². Using a continuation method, we also find stable artificial halo orbits which can be sustained for long integration times and require a reasonably small low-thrust acceleration 0.0593 mm/s².     

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.     

Correlation between Solar Flare Productivity and Photospheric Magnetic Field Properties II. Magnetic Gradient and Magnetic Shear

With 1353 vector magnetograms observed at Huairou Solar Observing Station (HSOS), a statistical analysis is made on the relationship among solar flares, magnetic gradient, and magnetic shear. The results suggest that flare productivity has positive correlations with the gradient and the shear, which can be well fitted by the Boltzmann sigmoidal function. In the vicinity of neutral lines, high gradient and strong shear are roughly coincident in time but barely in position. In addition, flare productivity is more sensitive to the length of neutral lines with strong gradient and shear (L _gs) than independently with strong gradient (L _g) or strong shear (L _s), which means that L _gs can be a better parameter for solar flare forecasting models. Finally, an algorithm to evaluate projection effects on the statistical results is proposed.     

The short‐period low‐mass binary system CC Com revisited

In this study we determined precise orbital and physical parameters of the very short‐period low‐mass contact binary system CC Com. The parameters are obtained by analysis of new CCD data combined with archival spectroscopic data. The physical parameters of the cool and hot components are derived as M_c = 0.717(14) M_⊙, M_h = 0.378(8) M_⊙, R_c = 0.708(12) R_⊙, R_h = 0.530(10) R_⊙, L_c = 0.138(12) L_⊙, and L_h = 0.085(7) L_⊙, respectively, and the distance of the system is estimated as 64(4) pc. The times of minima obtained in this study and with those published before enable us to calculate the mass transfer rate between the components which is 1.6 × 10^–8 M_⊙ yr^–1. Finally, we discuss the possible evolutionary scenario of CC Com (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)