Some new Runge-Kutta methods with interpolation properties and their application to the Magnetic-Binary Problem

Explicit Runge-Kutta methods provide a popular way to solve the initial value problem for a system of nonstiff ordinary differential equations. On the other hand, for these methods, there is no a natural way to approximate the solution at any point within a given integration step. Scaled Runge-Kutta methods have been developed recently which determine the solution of the differential system at non-mesh points of a given integration step. We propose some new such algorithms based upon well known explicit Runge-Kutta methods, and we verify their advantages by applying them to the Magnetic-Binary Problem.     

Local regularization of the Magnetic-Binary problem

The singularities in a Magnetic-Binary system are regularized separately by changing both the coordinates and the time. It is shown why, in this problem it is more efficient to relate the geometric transformation to the rescaling of time.     

Global regularization of the Magnetic-Binary problem

The author's aim is to achieve global regularization in the Magnetic-Binary problem by suitably transforming the state-time space of the system. The functions which perform the change of the physical time and the geometrical figures of the system, are connected by a special relation leaving the form of the equations of motion invariant. Additionally, a proposition for generalization of the process is discussed in an aspect as well, of how much such a regularization is profitable.     

Symmetric Periodic Orbits in the Anisotropic Schwarzschild-Type Problem

Studying the two-body problem associated to an anisotropic Schwarzschild-type field, Mioc et al. (2003) did not succeed in proving the existence or non-existence of periodic orbits. Here we answer this question in the affirmative. To do this, we start from two basic facts: (1) the potential generates a strong force in Gordon’s sense; (2) the vector field of the problem exhibits the symmetries S _i , , which form, along with the identity, an Abelian group of order 8 with three generators of order 2. Resorting to S ₂ and S ₃, in connection with variational methods (particularly the classical lower-semicontinuity method), we prove the existence of infinitely many S ₂- or S ₃-symmetric periodic solutions. The symmetries S ₂ and S ₃ constitute an indicator of the robustness of the classical isotropic Schwarzschild-type system to perturbations (as the anisotropy may be considered).     

Planar Symmetric Periodic Orbits in Four Dipole Problem

We have extend Stormer’s problem considering four magnetic dipoles in motion trying to justify the phenomena of extreme “orderlines” such as the ones observed in the rings of Saturn; the aim is to account the strength of the Lorentz forces estimating that the Lorentz field, co-acting with the gravity field of the planet, will limit the motion of all charged particles and small size grains with surface charges inside a layer of about 200 m thickness as that which is observed in the rings of Saturn. For this purpose our interest feast in the motion of charged particles with neglected mass where only electromagnetic forces accounted in comparison to the weakness of the Newtonian fields. This study is particularly difficult because in the regions we investigate these motions there is enormous three dimensional instability. Following the Poincare’s hypothesis that periodic solutions are ‘dense’ in the set of all solutions in Hamiltonian systems we try to calculate many families of periodic solutions and to study their stability. In this work we prove that in this environment charged particles can trace planar symmetric periodic orbits. We discuss these orbits in details and we give their symplectic relations using the Hamiltonian formulation which is related to the symplectic matrix. We apply numerical procedures to find families of these orbits and to study their stability. Moreover we give the bifurcations of these families with families of planar asymmetric periodic orbits and families of three dimensional symmetric periodic orbits.     

Central Configurations of the Symmetric Restricted 4-Body Problem

We consider the symmetric planar (3 + 1)-body problem with finite masses m ₁ = m ₂ = 1, m ₃ = µ and one small mass m ₄ = . We count the number of central configurations of the restricted case = 0, where the finite masses remain in an equilateral triangle configuration, by means of the bifurcation diagram with as the parameter. The diagram shows a folding bifurcation at a value consistent with that found numerically by Meyer [9] and it is shown that for small > 0 the bifurcation diagram persists, thus leading to an exact count of central configurations and a folding bifurcation for small m ₄ > 0.     

Dynamo action in the ionosphere and motions of the magnetospheric plasma I. Symmetric dynamo action

In order to envisage the circulation pattern of the magnetospheric plasma produced by the dynamo action in the ionosphere, the distribution of the dynamo-induced electrostatic field resulting from basic ionospheric wind systems is studied. It is then shown by use of Maeda's field distribution that there exists a remarkable large-scale circulation of the magnetospheric plasma, inward (earthward) on the evening side of the magnetosphere and outward on the morning side. This motion is comparable to the motion produced by the Earth's rotation and by zonal winds in the ionosphere. It is shown also that the electrostatic field can cause a considerable radial motion of some of the energetic particles in the radiation belt.     

Periodic Solutions for any Planar Symmetric Perturbation of the Kepler Problem

We consider perturbations of the Kepler problem that are symmetric with respect to the origin and admit a first integral of motion which is also symmetric with respect to the origin. It has been proved that each circular solution of the unperturbed problem gives rise to a periodic solution of the perturbed system.     

The equilibrium state in the Magnetic-Binary problem when the more massive primary is an oblate spheroid

The author's aim is to study in this paper the stationary state of the planar Magnetic-Binary problem by taking into consideration the oblateness of the more massive primary, and then to investigate the stability of motion about the equilibrium points by means of the characteristic polynomial of the linearised variational equations.     

Evolution of the General Solution of the Restricted Problem Covering Symmetric and Escape Solutions

The work presented in paper I (Papadakis, K.E., Goudas, C.L.: Astrophys. Space Sci. (2006)) is expanded here to cover the evolution of the approximate general solution of the restricted problem covering symmetric and escape solutions for values of μ in the interval [0, 0.5]. The work is purely numerical, although the available rich theoretical background permits the assertions that most of the theoretical issues related to the numerical treatment of the problem are known. The prime objective of this work is to apply the ‘Last Geometric Theorem of Poincaré’ (Birkhoff, G.D.: Trans. Amer. Math. Soc. 14, 14 (1913); Poincaré, H.: Rend. Cir. Mat. Palermo 33, 375 (1912)) and compute dense sets of axisymmetric periodic family curves covering the initial conditions space of bounded motions for a discrete set of values of the basic parameter μ spread along the entire interval of permissible values. The results obtained for each value of μ, tested for completeness, constitute an approximation of the general solution of the problem related to symmetric motions. The approximate general solution of the same problem related to asymmetric solutions, also computable by application of the same theorem (Poincaré-Birkhoff) is left for a future paper. A secondary objective is identification-computation of the compact space of escape motions of the problem also for selected values of the mass parameter μ. We first present the approximate general solution for the integrable case μ = 0 and then the approximate solution for the nonintegrable case μ = 10⁻³. We then proceed to presenting the approximate general solutions for the cases μ = 0.1, 0.2, 0.3, 0.4, and 0.5, in all cases building them in four phases, namely, presenting for each value of μ, first all family curves of symmetric periodic solutions that re-enter after 1 oscillation, then adding to it successively, the family curves that re-enter after 2 to 10 oscillations, after 11 to 30 oscillations, after 31 to 50 oscillations and, finally, after 51 to 100 oscillations. We identify in these solutions, considered as functions of the mass parameter μ, and at μ = 0 two failures of continuity, namely: 1. Integrals of motion, exempting the energy one, cease to exist for any infinitesimal positive value of μ. 2. Appearance of a split into two separate sub-domains in the originally (for μ = 0) unique space of bounded motions. The computed approximations of the general solution for all values of μ appear to fulfill the ‘completeness’ criterion inside properly selected sub-domains of the domain of bounded motions in the (x, C) plane, which means that these sub-domains are filled countably densely by periodic family curves, which form a laminar flow-line pattern. The family curves in this pattern may, or may not, be intersected by a ‘basic’ family curve segment of order from 1 up to 3. The isolated points generating asymptotic solutions resemble ‘sink’ points toward which dense sets of periodic family curves spiral. The points in the compact domain in the (x, C) plane resting outside the domain of bounded motions (μ = 0), including the gap between the two large sub-domains (μ > 0) created by the aforementioned split, generate escape motions. The gap between the two large sub-domains of bounded motions grows wider for growing μ. Also, a number of compact gaps that generate escape motions exist within the body of the two sub-domains of bounded motions. The approximate general solutions computed include symmetric, heteroclinic, asymptotic, collision and escape solutions, thus constituting one component of the full approximate general solution of the problem, the second and final component being that of asymmetric solutions.     

Repeat Ground Track Orbits of the Earth Tesseral Problem as Bifurcations of the Equatorial Family of Periodic Orbits

Orbits repeating their ground track on the surface of the earth are found to be members of periodic-orbit families (in a synodic frame) of the tesseral problem of the Earth artificial satellite. Families of repeat ground track orbits appear as vertical bifurcations of the equatorial family of periodic orbits, and they evolve from retrograde to direct motion throughout the 180 degrees of inclination. These bifurcations are always close to the resonances of the Earth's rotation rate and the mean motion of the orbiter.     

Large-scale motions in the sun

Large-scale solar motions comprise differential rotation (with latitudinal, and perhaps radial gradients), axially symmetric meridional motions, and possible asymmetric motions (giant convective cells or Rossby-type waves or both). These motions must be basic in any satisfactory theory of the changing pattern of solar magnetic fields and of the 22-yr cycle. In the present paper available data are discussed and, as far as possible, evaluated and explained.Rotational measurements are based on the changing positions of discrete features such as sunspots, on Doppler shifts, on geophysical changes and on statistical evaluation of the motions of diffuse objects. The first mentioned, comprising faculae, sunspots, K-corona (to latitudes 45°) and filaments, show agreement better than 0.7 %. A new formula for surface rotation _s , based on faculae and sunspot data, is _s = 14.52 – 2.48 sin² b – 2.51 sin⁶ b deg day^–1, where b is latitude, and validity may extend to about 70°. Errors in Doppler shift measurements and statistical treatments are discussed. There is evidence of a much slower coronal rate at high latitudes, and of a slower sub-surface rate at lower latitudes.Ordered meridional motions have been revealed by statistical investigations of the positions of spot groups, of spots and of filaments. All these results seem explicable in terms of an oscillating hydro-magnetic circulation in each hemisphere. These have both 11-yr and 22-yr components, and these periods are provided by a general dipole field of about one gauss, together with a pair of toroidal fields centred at latitudes ±16° and of average strength of order 10 G.Evidence of large-scale (perhaps 3 × 10⁵ km), irregular surface motions is provided by the distribution of surface magnetic flux, the motions of sunspots, and Doppler-shift observations; it is supported by Ward's theory of the equatorial acceleration. The possibility is suggested that these asymmetric motions also drive the oscillatory meridional motions.     

A Systematic Study of the Stability of Symmetric Periodic Orbits in the Planar, Circular, Restricted Three-Body Problem

We investigate symmetric periodic orbits in the framework of the planar, circular, restricted, three-body problem. Having fixed the mass of the primary equal to that of Jupiter, we determine the linear stability of a number of periodic orbits for different values of the eccentricity. A systematic study of internal resonances, with frequency p/q with 2p 9, 1 q 5 and 4/3 p/q 5, offers an overall picture of the stability character of inner orbits. For each resonance we compute the stability of the two possible periodic orbits. A similar analysis is performed for some external periodic orbits.Furthermore, we let the mass of the primary vary and we study the linear stability of the main resonances as a function of the eccentricity and of the mass of the primary. These results lead to interesting conclusions about the stability of exosolar planetary systems. In particular, we study the stability of Earth-like planets in the planetary systems HD168746, GI86, 47UMa,b and HD10697.     

Equatorial depletions in the 630.0 nm airglow at Vanimo

Using ground based airglow photometry, depletions in the 630.0 nm airglow were observed at Vanimo near the southern limb of the intertropical airglow arc. The results were compared with the more common properties of equatorial plasma bubbles such as depletion magnitude, cross-sectional size and East-West drift, with good agreement. In particular, airglow depletion depths ranged from 18 to 64% with a maximum loss in emission rate of 700 R (55%) on a night when the maximum recorded airglow was almost 1700 R. This corresponds to an electron density depletion of about 1.5 × 10¹² el m ^?3 observed near solar maximum. It is somewhat higher than values reported near solar minimum. The airglow depletions move eastward with velocities ranging from 90 to 140 ms ^?1. There is qualitative evidence of vertical motion and strong correlation with range type spread F.     

Stability of the Triangular Libration Points in the Unrestricted Planar Problem of a Symmetric Rigid Body and a Point Mass

In papers (Godziewski and Maciejewski, 1998a, b, 1999), we investigate unrestricted, planar problem of a dynamically symmetric rigid body and a sphere. Following the original statement of the problem by Kokoriev and Kirpichnikov (1988), we assume that the potential of the rigid body is approximated by the gravitational field of a dumb-bell. The model is described in terms of a 2D Hamiltonian depending on three parameters.In this paper, we investigate the stability of triangular equilibria permissible by the dynamics of the model, under the assumption of low-order resonances. We analyze all resonances of order smaller than four, and we examine the stability with application of theorems by Markeev and Sokolsky. These are the possible following cases: the non-diagonal resonance of the first order with two null characteristic frequencies (unstable); resonances of the first order with one nonzero frequency (diagonal and non-diagonal, stable and unstable); the second-order resonance, which is non-diagonal and stable, and the third-order resonance which is generically unstable, except for three points in the parameters' space, corresponding to stable equilibria.We discuss a perturbed version of Kokoriev and Kirpichnikov model, and we find that if the perturbation is small and depends on the coordinates only, the triangular equilibria persist, except if for the unperturbed equilibria the first-order resonance occurs. We show that the resonances of the order higher than two are also preserved if the perturbation acts.     

Doubly-symmetric motions in the elliptic problem

Poincaré's continuation method is applied to the elliptic restricted problem for the computation of four families of doubly symmetric three-dimensional periodic orbits emanating from similar orbits corresponding to zero eccentricity. The results are given in four tables and the orbits are characterized in regard to their stability.     

Grain motions in the solar nebula

Isotopic analyses of meteorites suggest the possibility that some interaction between supernova ejecta and grains occurred in the solar nebula. In particular, the dynamics of grain motions in the solar nebula can explain the observed mixing of nucleosynthetic components. The effect of a shock wave on the motions of grains are examined. A steady-state, plane shock propagating into a uniform region of gas and dust grains is followed by a zone of gas/grain slip, in which the grains are accelerated by drag forces from the pre-shock to the post-shock gas velocity, i.e. reducing the relative velocity between the gas and grains to zero. On the basis of these calculations, it is estimated that if grains carried the isotopic anomalies investigated by Lee, Papanastassoiu, and Wasserburg (1978), then those grains could be no bigger than 2×10^–4 cm in size. A scenario is suggested in which the sluggishness of grains provides a natural way to concentrate and mix the nucleosynthetic components carried by grains in the ejecta and in the solar nebula.Paper presented at the Conference on Protostars and Planets, held at the Planetary Science Institute, University of Arizona, Tucson, Arizona, between January 3 and 7, 1978.     

Wisp motions in the Crab nebula

We present the results of a CCD monitoring campaign of the continuum emission from the central region of the Crab nebula, amounting to 17 epochs spread over 3.5 years. The data provide clear evidence that the brightest wisps move outward from the pulsar at mildly relativistic velocities. This motion, combined with the shape of the wisps, supports the idea that they arise at a standing shock in an equatorial wind. The deprojected velocity of the wisps in the equatorial plane is c/3. We see only small changes in the so-called ‘thin wisps’ which leads us to suggest that these wisps may be the result of a back-flow from the shock in a toroidal cavity around the pulsar.     

Mass motions in the transition region

Following previous order of magnitude estimates (Poletto, 1980), the possibility that hot downflowing motions in the solar transition region could be ascribed to spicular matter returning to the chromosphere after being heated by compression, is more thoroughly investigated. The equations describing the one-dimensional non stationary motion of the spicular plasma during the heating process are analytically solved, and the temporal profiles of temperature, density and velocity are given for a set of representative situations. The results are finally compared with available data.