Horseshoes and nonintegrability in the restricted case of a spinless axisymmetric rigid body in a central gravitational field

The purpose of this paper is to study the motion of a spinless axisymmetric rigid body in a Newtonian field when we suppose the motion of the center of mass of the rigid body is on a Keplerian orbit. In this case the system can be reduced to a Hamiltonian system with configuration space of a two-dimensional sphere. We prove that the restricted planar motion is analytical nonintegrable and we find horseshoes due to the eccentricity of the orbit. In the caseI ₃/I ₁>4/3, we prove that the system on the sphere is also analytical nonintegrable.On leave from the Polytechnic Institute of Bucharest, Romania.     

Families of periodic orbits in the restricted four-body problem

In this paper, families of simple symmetric and non-symmetric periodic orbits in the restricted four-body problem are presented. Three bodies of masses m ₁, m ₂ and m ₃ (primaries) lie always at the apices of an equilateral triangle, while each moves in circle about the center of mass of the system fixed at the origin of the coordinate system. A massless fourth body is moving under the Newtonian gravitational attraction of the primaries. The fourth body does not affect the motion of the three bodies. We investigate the evolution of these families and we study their linear stability in three cases, i.e. when the three primary bodies are equal, when two primaries are equal and finally when we have three unequal masses. Series, with respect to the mass m ₃, of critical periodic orbits as well as horizontal and vertical-critical periodic orbits of each family and in any case of the mass parameters are also calculated.     

On Motions Near the Lagrange Equilibrium Point L 4 in the Case of Routh's Critical Mass Ratio

We deal with the planar restricted circular problem of three bodies. We study trajectories in a small neighborhood of the Lagrange equilibrium point L ₄ when mass ratio is close to Routh's value. In particular, we show that the case of proper degeneracy takes place and for most initial conditions trajectories are conditionally-periodic. We obtain an approximate representation of families of periodic solution emanating from the equilibrium point L ₄. We also show that in the case of instability of L ₄ the trajectories starting in a vicinity of L ₄ remain in a finite domain forever. We give an upper bound of this domain. To carry out our investigation, we analyze the dynamics of a general Hamiltonian system with two degrees of freedom in the case of 1 : 1 resonance in detail.This revised version was published online in October 2005 with corrections to the Cover Date.     

Homoclinic and heteroclinic orbits in the photogravitational restricted three-body problem

We study numerically the asymptotic homoclinic and heteroclinic orbits around the hyperbolic Lyapunov periodic orbits which emanate from Euler's critical points L ₁ and L ₂, in the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated to these Lyapunov orbits, are also presented. Poincaré surface of sections of these manifolds on appropriate planes and several homoclinic and heteroclinic orbits for the gravitational case as well as for varying radiation factor q ₁, are displayed. Homoclinic-homoclinic and homoclinic-heteroclinic-homoclinic chains which link the interior with the exterior Hill's regions, are illustrated. We adopt the Sun-Jupiter system and assume that only the larger primary radiates. It is found that for small deviations of its value from the gravitational case (q ₁ = 1), the radiation pressure exerts a significant impact on the Hill's regions and on these asymptotic orbits.     

Mass-to-luminosity ratio in binary galaxies

We have compared the observed distribution of the quantity log(V ₂ ² r_P) for a sample of 233 pairs of galaxies with Monte-Carlo simulations. From such an analysis we have derived an average mass-to-luminosity ratiosM/L _B =18±11. Our result is consistent with a linear increase of the mass with radius at least until distances of about 30 kpc.     

Photometric investigation of the evolved contact binary system DN Aur

We present here the photometric light curve analyses of the eclipsing binary star DN Aur. The CCD photometry, performed at the Behlen observatory using the 0.76 m automated telescope gave 646 individual data points inV andR bandpass filters. From this data we have determined a new epoch and an orbital period of 0.6168891 days. The published spectral classification is F3.The Wilson-Devinney model was used to derive the photometric solutions. DN Aur is a W UMa type contact binary system. The mass ratio, (q=m ₂/m ₁0.210, where star 2 is eclipsed at the primary minimum) suggests that the system has A-type configuration. The computed light curve has a third light of about 22 percent and a total eclipse in the secondary minimum. A solution with a cool spot on the secondary component is also found. We recommend spectroscopic study of DN Aur even though the light curve analysis show it to be a single line spectroscopic system. Generally contact systems of spectral type F3 have periods ranging from 0.25 to 0.5 days. The longer period of DN Aur suggests that it is an evolved contact system with case A mass transfer.     

Preprocessing of Vector Magnetograph Data for a Nonlinear Force-Free Magnetic Field Reconstruction

Knowledge regarding the coronal magnetic field is important for the understanding of many phenomena, like flares and coronal mass ejections. Because of the low plasma beta in the solar corona, the coronal magnetic field is often assumed to be force-free and we use photospheric vector magnetograph data to extrapolate the magnetic field into the corona with the help of a nonlinear force-free optimization code. Unfortunately, the measurements of the photospheric magnetic field contain inconsistencies and noise. In particular, the transversal components (say B _x and B _y) of current vector magnetographs have their uncertainties. Furthermore, the magnetic field in the photosphere is not necessarily force free and often not consistent with the assumption of a force-free field above the magnetogram. We develop a preprocessing procedure to drive the observed non–force-free data towards suitable boundary conditions for a force-free extrapolation. As a result, we get a data set which is as close as possible to the measured data and consistent with the force-free assumption.     

Magnetic fields and quasi-periodic oscillations of black hole radiation

Various relations are found between the key parameters of black holes and active galactic nuclei. Some have a statistical property, others follow from the theoretical consideration of the evolution of these objects. In this paper we use a recently discovered empirical relation between the characteristic frequency of quasi-periodic oscillations of radiation ν _br of black holes, their masses and matter accretion rates to determine the magnetic field strength B _H at the black hole event horizon. Since the characteristic frequency can be determined from observations, the use of a new relation for the estimations of magnetic field B _H can yield more definite results, since we are decreasing the number of the unknown or poorly-determined parameters of objects (it especially concerns the accretion rate Ṁ). The typical values which we have found are B _H ≃ 10⁸G for the stellar mass black holes, and B _H ≃ 10⁴G for the supermassive black holes. Besides, we demonstrate that if the linear polarization of an object is caused by the radiation of a magnetized accretion disk, then the degree of observable polarization is p ∼ ν _br ^−1/2.     

Tidal effects of a massive spheroidal halo on an inner spheroidal contracting visible body: Application to an evolutionary disk-galaxy model

The effects of the tidal interactions between two coaxial, homogeneous spheroids, one (the “Brigt Component”: B) completely embedded in the other (the “Dark Halo”: D), along a quasi-static contraction, are considered. The aim is to look how the dynamical properties and the final morphology of the B subsystem may be affected by the presence of the D component. Three initial configurations are considered: the quasi-spherical “Dark Halo” D coincides with the “Visible Component” B (case C); D is flatter than B but the two spheroids have the same semiminor axis (case N); no D component is assumed; the “visible” spheroid is single (case S). The application to an evolutionary disk-galaxy model is considered under some simple assumptions: i) in cases C and N the spheroidal halo is massive (mass ratio “dark”/“bright” about ten) and dissipationless. For a mass ratio like this, the tidal interaction of the B component over D turns to be negligible in the course of the contraction; adding to that the lack of dissipation, it appears plausible to take the D component as frozen along the evolution; ii) the degree of anisotropy and the angular momentum of B, J_B, are conserved. The conservation of J_B provides us the time-independent relationship among the key physical quantities and gives the possibility to draw the evolutionary tracks on the plane (axis ratio σ_B, semimajor axis a_B) without any explicit time-scale; iii) the global “star formation rate” is parametrized according to a simple “Schmidt power law” proportional to the square of gas density. At every step of the quasi-static contraction, the structure is determined by the tensor virial theorem extended to a double configuration. The model is very idealized, particularly because there are no available tidal gravitational terms other than for the case of two homogeneous. Nevertheless, the method based on the tensor virial appears powerful to gain insight into the correlations among the physical quantities involved and into their trend along the evolution. One of the main result is a clear indication of a leading role played by the axis ratio of the “Dark-Halo” component which might be, to the extent that this simple picture can be compared with a real galactic system, a possible new physical parameter to be added to mass and angular momentum for separating spirals from S0 galaxies.     

Bifurcation of Relative Equilibria in the Main Problem of Artificial Satellite Theory for a Prolate Body

In this paper, we study circular orbits of the J ₂ problem that are confined to constant-z planes. They correspond to fixed points of the dynamics in a meridian plane. It turns out that, in the case of a prolate body, such orbits can exist that are not equatorial and branch from the equatorial one through a saddle-center bifurcation. A closed-form parametrization of these branching solutions is given and the bifurcation is studied in detail. We show both theoretically and numerically that, close to the bifurcation point, quasi-periodic orbits are created, along with two families of reversible orbits that are homoclinic to each one of them.     

Integration theory for the restricted problem of three axisymmetric bodies

In this paper the circular planar restricted problem of three axisymmetric ellipsoids S _i (i = 1, 2, 3), such that their equatorial planes coincide with the orbital plane of the three centres of masses, be considered. The equations of motion of infinitesimal body S ₃ be obtained in the polar coordinates. Using iteration approach we have given an approximation for another integral, which independent of the Jacobian integral, in the case of P-type orbits (near circular orbits surrounding both primaries).     

Symmetric and Asymmetric Librations in Planetary and Satellite Systems at the 2/1 Resonance

Families of asymmetric periodic orbits at the 2/1 resonance are computed for different mass ratios. The existence of the asymmetric families depends on the ratio of the planetary (or satellite) masses. As models we used the Io-Europa system of the satellites of Jupiter for the case m₁>m₂, the system HD82943 for the new masses, for the case m₁=m₂ and the same system HD82943 for the values of the masses m₁<m₂ given in previous work. In the case m₁≥ m₂ there is a family of asymmetric orbits that bifurcates from a family of symmetric periodic orbits, but there exist also an asymmetric family that is independent of the symmetric families. In the case m₁<m₂ all the asymmetric families are independent from the symmetric families. In many cases the asymmetry, as measured by and by the mean anomaly M of the outer planet when the inner planet is at perihelion, is very large. The stability of these asymmetric families has been studied and it is found that there exist large regions in phase space where we have stable asymmetric librations. It is also shown that the asymmetry is a stabilizing factor. A shift from asymmetry to symmetry, other elements being the same, may destabilize the system.     

Constraining the relative inclinations of the planets B and C of the millisecond pulsar PSR B1257+12

We investigate on the relative inclination of the planets B and C orbiting the pulsar PSR B1257+12. First, we show that the third Kepler’s law does represent an adequate model for the orbital periods P of the planets, because other Newtonian and Einsteinian corrections are orders of magnitude smaller than the accuracy in measuring P _B/C. Then, on the basis of available timing data, we determine the ratio sin i _C/ sin i _B = 0.92±0.05 of the orbital inclinations i _B and i _C independently of the pulsar’s mass M. It turns out that coplanarity of the orbits of B and C would imply a violation of the equivalence principle. Adopting a pulsar mass range 1 ≲ M ≲ 3, in solar masses (supported by present-day theoretical and observational bounds for pulsar’s masses), both face-on and edge-on orbital configurations for the orbits of the two planets are ruled out; the acceptable inclinations for B span the range 36 deg ≲ i _B ≲ 66 deg, with a corresponding relative inclination range 6 deg ≲ (i _C − i _B) ≲ 13 deg.     

Rosette Central Configurations, Degenerate Central Configurations and Bifurcations

In this paper we find a class of new degenerate central configurations and bifurcations in the Newtonian n-body problem. In particular we analyze the Rosette central configurations, namely a coplanar configuration where n particles of mass m₁ lie at the vertices of a regular n-gon, n particles of mass m₂ lie at the vertices of another n-gon concentric with the first, but rotated of an angle π /n, and an additional particle of mass m₀ lies at the center of mass of the system. This system admits two mass parameters μ = m₀/m₁ and ε = m₂/m₁. We show that, as μ varies, if n > 3, there is a degenerate central configuration and a bifurcation for every ε > 0, while if n = 3 there is a bifurcation only for some values of ε.     

The Crust Properties and Cooling of Strange Stars

We investigate the influence of the following parameters on the crust properties of strange stars: the strange quark mass (m _s), the strong coupling constant (α_c) and the vacuum energy density (B). It is found that the mass density at the crust base of strange stars cannot reach the neutron drip density. For a conventional parameter set of m _s=200 MeV, B ^1/4 = 145 MeV and α_c = 0.3, the maximum density at the crust base of a typical strange star is only 5.5 × 10¹⁰ gcm^-3, and correspondingly the maximum crust mass is 1.4 ×10^-6 M_⊙. Subsequently, we present the thermal structure and the cooling behavior of strange stars with crusts of different thickness, and under different diquark pairing gaps. Our work might provide important clues for distinguishing strange stars from neutron stars.     

The structure of chaos in a potential without escapes

We study the structure of chaos in a simple Hamiltonian system that does no have an escape energy. This system has 5 main periodic orbits that are represented on the surface of section by the points (1)O(0,0), (2)C ₁,C ₂(±y _c, 0), (3)B ₁,B ₂(O,±1) and (4) the boundary . The periodic orbits (1) and (4) have infinite transitions from stability (S) to instability (U) and vice-versa; the transition values of are given by simple approximate formulae. At every transitionS U a set of 4 asymptotic curves is formed atO. For larger the size and the oscillations of these curves grow until they destroy the closed invariant curves that surroundO, and they intersect the asymptotic curves of the orbitsC ₁,C ₂ at infinite heteroclinic points. At every transitionU S these asymptotic curves are duplicated and they start at two unstable invariant points bifurcating fromO. At the transition itself the asymptotic curves fromO are tangent to each other. The areas of the lobes fromO increase with ; these lobes increase even afterO becomes stable again. The asymptotic curves of the unstable periodic orbits follow certain rules. Whenever there are heteroclinic points the asymptotic curves of one unstable orbit approach the asymptotic curves of another unstable orbit in a definite way. Finally we study the tangencies and the spirals formed by the asymptotic curves of the orbitsB ₁,B ₂. We find indications that the number of spiral rotations tends to infinity as . Therefore new tangencies between the asymptotic curves appear for arbitrarily large . As a consequence there are infinite new families of stable periodic orbits that appear for arbitrarily large .     

Hydromagnetic stability of some non-dissipative flows subject to non-axisymmetric disturbances

We study the linear stability of nondissipative flow of an electrically conducting fluid subject to non-axisymmetric disturbances in the following cases: (i) the radial flow of an incompressible fluid between two concentric porous circular cylinders in the presence of a radial magnetic field and (ii) axial flow of a compressible fluid between two concentric circular cylinders permeated by a helical magnetic field (0,B ₀(r),B _0z) in a cylindrical coordinate system. It is shown that in case (i), the flow is stable if the Alfvén velocity based on the undisturbed radial magnetic field exceeds the radial velocity due to suction or injection at the cylinder surfaces. In case (ii), it is found that under certain conditions the complex wave speed for an unstable mode lies within a circle of diameterW _max-W _min, whereW _max andW _min are the maximum and minimum values of the axial velocity in the flow region. In the presence of a purely axial magnetic field, however, the complex wave speed for an unstable mode always lies within the above circle.     

1990 photometry of a small amplitude W UMa system LS Delphini

DifferentialUBV observations, carried out in 1990 observing season, of a small amplitude (0 _. ^m 15 inV andB) W UMa system LS Del = HD 199497 are presented. Wavelength-dependent light variations from cycle to cycle indicate that the system is in a very active phase, probably due to magnetic flare activity or mass transfer in the system. An analysis of the minima times indicate a probable secular increase of the photometric period which requires a mass transfer from less massive to more massive component. If this is true then the reverse-algol model by Liuet al. (1988) for this system would not be valid.     

Bimodal brightness oscillations in models of young binary systems

We consider a model for the cyclic activity of young binary stars that accrete matter from the remnants of a protostellar cloud. If the orbit of such a binary system is inclined at a small angle to the line of sight, then the streams of matter and the density waves excited in the circumbinary disk can screen the primary component of the binary from the observer. To study these phenomena by the SPH (smoothed particle hydrodynamics) method, we have computed grids of hydrodynamic models for binary systems based on which we have constructed the light curves as a function of the orbital phase. The main emphasis is on investigating the properties of the brightness oscillations. Therefore, the model parameters were varied within the following ranges: the component mass ratio q = M ₂: M ₁ = 0.2–0.5 and the eccentricity = 0–0.7. The parameter that defined the binary viscosity was also varied. We adopted optical grain characteristics typical of circumstellar dust. Our computations have shown that bimodal oscillations are excited in binaries with eccentric orbits, provided that the binary components do not differ too much in mass. In this case, the ratios of the periods and amplitudes of the bimodal oscillations and their shape depend strongly on the inclination of the binary plane and its orientation relative to the observer. Our analysis shows that the computed light curves can be used in interpreting the cyclic activity of UX Ori stars.     

The Variation of Solar Wind Correlation Lengths Over Three Solar Cycles

We present the results of a study of solar wind velocity and magnetic field correlation lengths over the last 35 years. The correlation length of the magnetic field magnitude λ _|B| increases on average by a factor of two at solar maxima compared to solar minima. The correlation lengths of the components of the magnetic field l_BXYZ\lambda_{B_{XYZ}} and of the velocity l_VYZ\lambda_{V_{YZ}} do not show this change and have similar values, indicating a continual turbulent correlation length of around 1.4×10⁶ km. We conclude that a linear relation between λ _|B|, VB ², and Kp suggests that the former is related to the total magnetic energy in the solar wind and an estimate of the average size of geoeffective structures, which is, in turn, proportional to VB ². By looking at the distribution of daily correlation lengths we show that the solar minimum values of λ _|B| correspond to the turbulent outer scale. A tail of larger λ _|B| values is present at solar maximum causing the increase in mean value.