The equilibrium configurations of the restricted problem of 2+2 triaxial rigid bodies

The restricted 2+2 body problem is considered. The infinitesimal masses are replaced by triaxial rigid bodies and the equations of motion are derived in Lagrange form. Subsequently, the equilibrium solutions for the rotational and translational motion of the bodies are detected. These solutions are conveniently classified in groups according to the several combinations which are possible between the translational equilibria and the constant orientations of the bodies.     

Equilibrium solutions of the restricted problem of 2 + 2 axisymmetric rigid bodies

The restricted problem of 2 + 2 homogenous axisymmetric ellipsoids such that their equatorial planes coincide with the orbital plane of the centers of mass is considered. The equilibrium solutions of this problem are shown to exist. Six of these solutions are located about the collinear points of the restricted problem of three axisymmetric ellipsoids. A special case of this problem is studied and sixteen solutions are found in the neighborhood of the triangular Lagrangian points.     

The onset of chaotic motion in the restricted problem of three bodies

A full characterization of a nonintegrable dynamical system requires an investigation into the chaotic properties of that system. One such system, the restricted problem of three bodies, has been studied for over two centuries, yet few studies have examined the chaotic nature of some ot its trajectories. This paper examines and classifies the onset of chaotic motion in the restricted three-body problem through the use of Poincaré surfaces of section, Liapunov characteristic numbers, power spectral density analysis and a newly developed technique called numerical irreversibility. The chaotic motion is found to be intermittent and becomes first evident when the Jacobian constant is slightly higher thanC ₂.     

Stability and regions of motion in the restricted three-body problem when both the primaries are finite straight segments

In this paper we have studied the locations and stability of the Lagrangian equilibrium points in the restricted three-body problem under the assumption that both the primaries are finite straight segments. We have found that the triangular equilibrium points are conditional stable for 0<μ<μ _c , and unstable in the range μ _c <μ≤1/2, where μ is the mass ratio. The critical mass ratio μ _c depends on the lengths of the segments and it is observed that the range of μ _c increases when compared with the classical case. The collinear equilibrium points are unstable for all values of μ. We have also studied the regions of motion of the infinitesimal mass. It has been observed that the Jacobian constant decreases when compared with the classical restricted three-body problem for a fixed value of μ and lengths l ₁ and l ₂ of the segments. Beside this we have found the numerical values for the position of the collinear and triangular equilibrium points in the case of some asteroids systems: (i) 216 Kleopatra-951 Gaspara, (ii) 9 Metis-433 Eros, (iii) 22 Kalliope-243 Ida and checked the linear stability of stationary solutions of these asteroids systems.     

Equilibrium points and hill’s regions in the 1+2 and 2+2 problem

For the 1+2 and 2+2 problems, we obtain the equilibrium points, their stability, and the topology of the constant energy manifold.     

An approximate integral and solution for eccentric planetary type orbits in the restricted problem of three bodies

The circular restricted problem of three bodies is investigated analytically with respect to the problem of deriving a second integral of motion besides the well known Jacobian Integral. The second integral is searched for as a correction the angular momentum integral valid in the two body case. A partial differential equation equivalent to the problem is derived and solved approximately by an asymptotic Fourier method assuming either sufficiently small values for the dimensionless mass parameter or sufficiently large distances from the barycentre. The solution of the partial equation then leads to a function of the coordinates, velocities and time being nearly constant, which means that its variation with time is about 40–300 times less than that of the pure angular momentum. By averaging over the remaining fluctuating part of the quasi-integral we are able to integrate the first order equations using a renormalization transformation. This leads to an explicit expression for the approximate solution of the circular problem which describes the motion of the third body orbiting both primaries with nonvanishing initial eccentricity (eccentric planetary type orbits). One of the main results is an explicit formula for the frequency of the perihelion motion of the third body which depends on the mass parameter, the initial distance of the third body from the barycentre and the initial eccentricity. Finally we study orbits of the P-Type, being defined as solutions of the restricted problem with circular initial conditions (vanishing initial eccentricity).     

Modelling the chromosphere and transition region of ε Eri (K2 V)

Measurements of ultraviolet line fluxes from Space Telescope Imaging Spectrograph and Far-Ultraviolet Spectroscopic Explorer spectra of the K2-dwarf ε Eri are reported. These are used to develop new emission measure distributions and semi-empirical atmospheric models for the chromosphere and lower transition region of the star. These models are the most detailed constructed to date for a main-sequence star other than the Sun. New ionization balance calculations, which account for the effect of finite density on dielectronic recombination rates, are presented for carbon, nitrogen, oxygen and silicon. The results of these calculations are significantly different from the standard Arnaud & Rothenflug ion balance, particularly for alkali-like ions. The new atmospheric models are used to place constraints on possible first ionization potential (FIP)-related abundance variations in the lower atmosphere and to discuss limitations of single-component models for the interpretation of certain optically thick line fluxes.     

Simultaneous optical polarimetry and X-ray observations of the magnetic CV CP Tuc (AX J2315−592)

CP Tuc (AX J2315–592) shows a dip in X-rays which lasts for approximately half the binary orbit and is deeper in soft X-rays compared with hard X-rays. It has been proposed that this dip is due to the accretion stream obscuring the accretion region from view. If CP Tuc were a polar, as has been suggested, then the length of such a dip would make it unique amongst polars since in those polars in which a dip is seen in hard X-rays the dip lasts for only 0.1 of the orbit. We present optical polarimetry and RXTE observations of CP Tuc which show circular polarization levels of ∼10 per cent and find evidence for only one photometric period. These data confirm CP Tuc as a polar. Our modelling of the polarization data implies that the X-ray dip is due to the bulk of the primary accretion region being self-eclipsed by the white dwarf. The energy dependence of the dip is due to a combination of this self-eclipse and also the presence of an X-ray temperature gradient over the primary accretion region.