On the theoretical possibility of the libration cloud

We show within the framework of the restricted three body problem that

1. Only in the immediate neighbourhood of the Lagrangian pointsL ₄ andL ₅ the distribution of a cloud of particles tends to become uniform under the influence of random stochastic perturbations. No consequences can be derived from this fact for a tendency of dispersion of clouds librating at arbitrary distances around the Lagrangian points.
2. From an elementary inspection of the Jacobi integral we cannot conclude that mutual completely inelastic collisions tend to drive the particles away from the vicinity of the libration points.

Finally the motion of a particle within the libration cloud, approximated as a resisting medium, is briefly examined.     

Effect of perturbations in Coriolis and centrifugal forces on the stability of libration points in the restricted problem

The location and the stability of the libration points in the restricted problem have been studied when small perturbation and are given to the Coriolis and the centrifugal forces respectively. It is seen that the pointsL ₄ andL ₅ form nearly equilateral triangles with the primaries and the pointsL ₁,L ₂,L ₃ remain collinear. It is further observed that for the pointsL ₄ andL ₅, the range of stability increases or decreases depending upon whether the point (, ) lies in one or the other of the two parts in which the (, ) plane is divided by the line 36-19=0 and the stability of the collinear points is not influenced by the perturbations and they remain unstable.     

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.     

Stability of L4 and L5 against radiation pressure

The restricted three-body problem is generalized with the inclusion of solar radiation pressure. For small particles (typically 1 m to 1 mm) the familiar equilibrium triangular points L₄ and L₅ no longer exist. However libration orbits are not completely destroyed, although an effect of resonance causes their amplitude to be very large, for a particle initially at rest at either of the triangular point. Finally the results of a study of the linearized equations of motion, supplemented by a numerical integration, rule out the possibility of an accumulation of dust at the Earth-Moon lagrangian triangular points.     

Global Bifurcations of Periodic Solutions of the Restricted Three Body Problem

We describe global bifurcations from the libration points of non-stationary periodic solutions of the restricted three body problem. We show that the only admissible continua of non-stationary periodic solutions of the planar restricted three body problem, bifurcating from the libration points, can be the short-period families bifurcating from the Lagrange equilibria L ₄, L ₅. We classify admissible continua and show that there are possible exactly six admissible continua of non-stationary periodic solutions of the planar restricted three body problem. We also characterize admissible continua of non-stationary periodic solutions of the spatial restricted three body problem. Moreover, we combine our results with the Déprit and Henrard conjectures (see [8]), concerning families of periodic solutions of the planar restricted three body problem, and show that they can be formulated in a stronger way. As the main tool we use degree theory for SO(2)-equivariant gradient maps defined by the second author in [25].This revised version was published online in October 2005 with corrections to the Cover Date.     

On the relation between solar-flare gamma-ray emission and proton escape into interplanetary space

It is shown that escaping of solar flare energetic protons into interplanetary space as well as their relation to the flare gamma-ray emission depend on the parameter = 8p/B ₀ ² , where p is the pressure of hot plasma and energetic particles and B ₀ is the magnetic field in a flaring loop. If 1, the bulk of the energetic protons escape to the loss cone because of diffusion due to small-scale Alfvén-wave turbulence, and precipitate into the footpoints of the flaring loop. The flare then produces intense gamma-ray line emission and a weak flux of high energy protons in interplanetary space. If >_*0.3-1.0, then fast eruption of hot plasma and energetic particles out of the flaring loop occurs, this being due to the flute instability or magnetic-field-plasma nonequilibrium. The flare then produces a comparatively weak gamma-radiation and rather intense proton fluxes in interplanetary space. We predict a modulation of the solar flare gamma-ray line emission with a period 1 s during the impulsive phase that is due to the MHD-oscillations of the energy release volume. The time lag of the gamma-ray peaks with respect to the hard X-ray peaks during a simultaneous acceleration of electrons and protons can be understood in terms of strong diffusion.     

Almost rectilinear halo orbits

Numerical studies over the entire range of mass-ratios in the circular restricted 3-body problem have revealed the existence of families of three-dimensional halo periodic orbits emanating from the general vicinity of any of the 3 collinear Lagrangian libration points. Following a family towards the nearer primary leads, in 2 different cases, to thin, almost rectilinear, orbits aligned essentially perpendicular to the plane of motion of the primaries. (i) If the nearer primary is much more massive than the further, these thin L₃-family halo orbits are analyzed by looking at the in-plane components of the small osculating angular momentum relative to the larger primary and at the small in-plane components of the osculating Laplace eccentricity vector. The analysis is carried either to 1st or 2nd order in these 4 small quantities, and the resulting orbits and their stability are compared with those obtained by a regularized numerical integration. (ii) If the nearer primary is much less massive than the further, the thin L₁-family and L₂-family halo orbits are analyzed to 1st order in these same 4 small quantities with an independent variable related to the one-dimensional approximate motion. The resulting orbits and their stability are again compared with those obtained by numerical integration.     

A complete model of the propagation of solar-flare cosmic rays

A two-stage model of the propagation of 1–50 MeV solar-flare cosmic rays is presented. The first stage consists of a thin spherical shell of radius r _a near the Sun which feeds particles into interplanetary space (the second stage) where they propagate along the Archimedean mean interplanetary magnetic field under the influences of anisotropic diffusion, convection, and energy changes. To calculate the time dependence at a fixed point in space, account is taken of the corotation of flux tubes past the observer.It is shown that the well-known east-west effect of the time-to-maximum cannot be obtained if the injection from the first stage is impulsive and thus a time and longitude dependent release for the second stage is essential. This is achieved by treating the first stage as a thin, spherical, diffusing shell of radius r _a with diffusion coefficient _s, from which particles leak into interplanetary space at a rate determined by the leakage coefficient .With this model we are able to reproduce simultaneously four principal features of solar events observed at r = 1 AU: (i) the east-west effect, i.e. the time-to-maximum as a function of flare longitude; (ii) the three phases of the anisotropy vector variation; (iii) the time-to-convective-phase as a function of flare longitude; and (iv) the longitudinal distribution of the differential intensity. Our best estimates of the parameters of the near-Sun propagation are that 0.01 hr^–1 _s/r _a ² 0.02 hr^–1 and 1/15 hr^–1 1/10 hr^–1. For the interplanetary propagation we estimate /V - 1.2AU with , the effective cosmic-ray diffusion coefficient and V, the solar-wind speed.     

Geometrical dynamics: A new approach to periodic orbits aroundL 4

Geometrical dynamics is the study of the geometry of the orbits in configuration space of a dynamical system without reference to the system's motion in time.Generalized coordinates for the circular restricted problem of three bodies are taken as polar coordinatesr, centered at the triangular libration pointL ₄. A time-independent nonlinear second order ordinary differential equation forr as a function of is derived. Approximations to periodic solutions are obtained by perturbations and Fourier series.     

‘Dead’ pulsars: Cosmic-ray sources

Dead pulsars outnumber live pulsars by a factor of 10⁴. It is estimated that there are 3×10⁹ of them in our Galaxy. The exospheres of the atmosphere of dead pulsars are characterised by cosmic-ray energies per particle, as the result of accretion of cold particles from interstellar space. Velocities of particles in the exosphere tend to be Maxwellianised by collisions there. The temperature of the exosphere from which particles escape is of the order of 10¹² K while the temperature of the photosphere closer to the surface of the pulsar is of the order of 10⁷ K. Collisions in the exosphere result in Jeans's type escape of cosmic rays with GeV energies at infinity. Two braod ranges of conditions for the exospheres are considered (a) with no magnetic fields involved, and (b) with magnetic fields. Similar conclusions are reached regarding the escape of cosmic rays. Conditions are delineated such that the exospheres of dead pulsars might be major sources of cosmic rays.Paper dedicated to Professor Hannes Alfvén on the occasion of his 80th birthday, 30 May 1988.     

Theβ index in cool stars

If we follow recent work and in order to extend theuvby photometric calibrations to spectral types later than G0, we present an attempt to use the combineduvby and systems for stars in the range G5-K7 and luminosity classes V to III.The behaviour in the MK-, (b–y)– and cl– diagrams of the 200 stars, only good spectroscopic data being considered, suggests the usefulness of the index as an independent parameter for late-type Main-Sequence stars, following in a natural way the general trend defined by Crawford for F- and G-type stars.Paper presented at the 11th European Regional Astronomical Meetings of the IAU on New Windows to the Universe, held 3–8 July, 1989, Tenerife, Canary Islands, Spain     

Time-dependent Green's functions of the cosmic ray equation of transport

Two spherically symmetric time-dependent Green's functions of the equation of transport for cosmic rays in the interplanetary region are derived by transform techniques. The solar wind velocity is assumed radial and of constant speedV. In the first model the radial diffusion coefficient =₀ r (₀ constant), and in the second solution =₀= constant. The solutions are for monoenergetic, impulsive release of particles from a fixed heliocentric radius. Integration of the solutions over timet, fromt=0 tot=, gives the steady-state Green's functions obtained previously.     

Libration effects for retrograde satellites in the restricted three-body problem

Numerical explorations of the restricted problem have shown that for stable large nonperiodic retrograde satellite orbits, the motion can be decomposed into a fast reference motion and a slow libration aroundB ₂ We study here this libration in the circular plane Hill's case, for which the reference motion is elliptic. We establish the equations of motion for the coordinates of the centre of this ellipse. We find two integrals of motion: the first is the semi-major axis of the ellipse; the second is essentially Jacobi's integral, translated into the new coordinates. We give a formula for the period of the libration and we find its limiting value for small libration amplitudes. A numerical verification gives very good agreement for all these results.     

Non-linear stability around the triangular libration points

The configuration space around the triangular libration points in the Earth-Moon system is partitioned according to the stability of the motion. The regions aroundL ₄ andL ₅ are established where particles placed with zero initial velocity will librate. The complexity of the partitioning is revealed.     

Stability of the triangular libration points of the circular restricted problem in the presence of resonances

The stability of triangular libration points, when the bigger primary is a source of radiation and the smaller primary is an oblate spheroid. has been investigated in the resonance cases ₁ = 2₂ and ₁ = 3₂. The motion is unstable for all the values of parameters q and A when ₁ = 2₂ and the motion is unstable and stable depending upon the values of the parameters q and A when ₁ = 3₂. Here q is the radiation parameter and A is the oblateness parameter.     

On the stability of the triangular libration points for the photogravitational circular restricted problem of three bodies under the resonances of the third and the fourth order

We investigate the stability of the triangular libration points when both the attracting bodies are radiating under the resonance conditions ₁ = 2₂ and 3₂.     

Three-dimensional,periodic, ‘halo’ orbits

A largely numerical study was made of families of three-dimensional, periodic, halo orbits near the collinear libration points in the restricted three-body problem. Families extend from each of the libration points to the nearest primary. They appear to exist for all values of the mass ratio , from 0 to 1. More importantly, most of the families contain a range of stable orbits. Only near L₁, the libration point between the two primaries, are there no stable orbits for certain values of . In that case the stable range decreases with increasing , until it disappears at =0.0573. Near the other libration points, stable orbits exist for all mass ratios investigated between 0 and 1. In addition, the orbits increase in size with increasing .     

Circumstellar matter in the symbiotic binary EG and

We present a quantitative model of a proposed common slowly expanding envelope corotating with a symbiotic binary system. The geometry of the envelope is defined by the equipotential surface containing the Lagrangian pointL ₂. The model was applied to the symbiotic binary EG And. The orbital inclination,i = 45°, and mass ratio of the cool to the hot component,q = 2.8–3.0, are determined. The nature of the primary as well as the secondary minima in the light curves, development of the line profiles, radial velocities and equivalent widths of the H line through an orbital revolution are explained by this model.     

Processing of cometary grains at the nucleus surface

Cometary material inevitably undergoes chemical changes before and on leaving the nucleus. In seeking to explain comets as the origin of many IDPs (interplanetary dust particles), an understanding of potential surface chemistry is vital. Grains are formed and transformed at the nucleus surface; much of the cometary volatiles may arise from the organic material. In cometary near-surface permafrost, one expects cryogenic chemistry with crystal growth and isotope. This could be the hydrous environment where IDPs form. Seasonal and geographic variations imply a range of environmental conditions and surface evolution. Interplanetary dust impacts and electrostatic forces also have roles in generating cometary dust. The absence of predicted cometary dust envelopes is compatible with the wide range of particle structures and compositions. Study of IDPs would distinguish between this model and alternatives that see comets as aggregates of core-mantle grains built in interstellar clouds.     

Effect of Stellar Wind and Poynting–Robertson Drag on Photogravitational Elliptic Restricted Three Body Problem

The existence and linear stability of the planar equilibrium points for photogravitational elliptical restricted three body problem is investigated in this paper. Assuming that the primaries, one of which is radiating are rotating in an elliptical orbit around their common center of mass. The effect of the radiation pressure, forces due to stellar wind and Poynting–Robertson drag on the dust particles are considered. The location of the five equilibrium points are found using analytical methods. It is observed that the collinear equilibrium points L₁, L₂ and L₃ do not lie on the line joining the primaries but are shifted along the y-coordinate. The instability of the libration points due to the presence of the drag forces is demonstrated by Lyapunov’s first method of stability.