‘Similar’ coordinate systems and the Roche geometry: application

A new equivalence relation, named relation of ‘similarity’ is defined and applied in the restricted three-body problem. Using this relation, a new class of trajectories (named ‘similar’ trajectories) are obtained; they have the theoretical role to give us new details in the restricted three-body problem. The ‘similar’ coordinate systems allow us in addition to obtain a unitary and an elegant demonstration of some analytical relations in the Roche geometry. As an example, some analytical relations published by Seidov (in Astrophys. J. 603:283, 2004) are demonstrated.     

Global bifurcation of planar and spatial periodic solutions in the restricted <Emphasis Type="Italic">n</Emphasis>-body problem

The paper deals with the study of a satellite attracted by n primary bodies, which form a relative equilibrium. We use orthogonal degree to prove global bifurcation of planar and spatial periodic solutions from the equilibria of the satellite. In particular, we analyze the restricted three body problem and the problem of a satellite attracted by the Maxwell’s ring relative equilibrium.     

On the possible fractal nature of blazar light curves

Optical light curves of three blazars are analyzed by Hurst’s method of normalized range. It is shown that Hurst’s empirical relationship is satisfied for these curves, in accordance with which the Hurst parameters are found for each curve. Assuming that blazar light curves have self-affinity, they determine the fractal dimensionality of the curves to be D ≈ 1.1. Translated from Astrofizika, Vol. 40, No. 3, pp. 341–348, August, 1997.     

Wisdom system: Dynamics in the adiabatic approximation

A simple approximate model of the asteroid dynamics near the 3:1 mean–motion resonance with Jupiter can be described by a Hamiltonian system with two degrees of freedom. The phase variables of this system evolve at different rates and can be subdivided into the ‘fast’ and ‘slow’ ones. Using the averaging technique, wisdom obtained the evolutionary equations which allow to study the long-term behavior of the slow variables. The dynamic system described by the averaged equations will be called the ‘Wisdom system’ below. The investigation of the, wisdom system properties allows us to present detailed classification of the slow variables’ evolution paths. The validity of the averaged equations is closely connected with the conservation of the approximate integral (adiabatic invariant) possessed by the original system. Qualitative changes in the behavior of the fast variables cause the violations of the adiabatic invariance. As a result the adiabatic chaos phenomenon takes place. Our analysis reveals numerous stable periodic trajectories in the region of the adiabatic chaos.     

On particle trajectories in restricted 3-body problem including the effect of radiation: Sun-Jupiter system

In this work, we have simulated orbits of a particle moving in gravitational field of the Sun-Jupiter system. The effect of solar radiation pressure, including Poynting Robertson drag, on the evolution of particle orbits in phase space have been studied for different values of the parameter β ₁ (the ratio of radiation to gravitational force) and initial conditions. Characteristics of various computed trajectories have been studied using wavelet transform (WT), Fourier transform (FT) and Poincare surface of section method. We use wavelet analysis to identify transitions of a trajectory in time-frequency plane and further apply it to classify it as regular or chaotic in phase space. Unlike the Fourier transform method (FT), we observe that the wavelet transform (WT) also provides a basis to identify ‘sticky’ trajectories in the present dynamical system.     

Mutual phenomena of Jovian satellites

Results of the observations of mutual eclipses of Galilean satellites observed from the Vainu Bappu Observatory during 1985 are presented. Theoretical models assuming a uniform disc, Lambert’s law and Lommel-Seeliger’s law describing the scattering characteristics of the surface of the eclipsed satellite were used to fit the observations. Light curves of the 1E2 event on 1985 September 24 and the 3E1 event on 1985 October 24 observed from VBO and published light curves of the 1E2 event on 1985 September 14, the 3E1 event on 1985 September 26 and the 2E1 event on 1985 October 28 (Arlotet al 1989) were fitted with theoretical light curves using Marquardt’s algorithm. The best fitting was obtained using Lommel-Seeliger’s law to describe the scattering over the surface of Io and Europa. During the fitting, a parameter^δxshift which shifts the theoretical light curve along the direction of relative motion of the eclipsed satellite with respect to the shadow centre, on the sky plane (as seen from the Sun) was determined along with the impact parameter. In absence of other sources like prominent surface features or non perfect sky conditions which could lead to asymmetric light curves,^δxshift would be a measure of the phase correction (Aksnes, Franklin & Magnusson 1986) with an accuracy as that of the midtime. Heliocentric Δα cos (δ) and gDδ at mid times derived from fitted impact parameters are reported     

A numerical study of the orbits of second species of the planar circular RTBP

We present a numerical study of the set of orbits of the planar circular restricted three body problem which undergo consecutive close encounters with the small primary, or orbits of second species. The value of the Jacobi constant is fixed, and we restrict the study to consecutive close encounters which occur within a maximal time interval. With these restrictions, the full set of orbits of second species is found numerically from the intersections of the stable and unstable manifolds of the collision singularity on the surface of section that corresponds to passage through the pericentre. A ‘skeleton’ of this set of curves can be computed from the solutions of the two-body problem. The set of intersection points found in this limit corresponds to the S-arcs and T-arcs of Hénon’s classification which verify the energy and time constraints, and can be used to construct an alphabet to describe the orbits of second species. We give numerical evidence for the existence of a shift on this alphabet that describes all the orbits with infinitely many close encounters with the small primary, and sketch a proof of the symbolic dynamics. In particular, we find periodic orbits that combine S-type and T-type quasi-homoclinic arcs.     

The use of invariant manifolds for transfers between unstable periodic orbits of different energies

Techniques from dynamical systems theory have been applied to the construction of transfers between unstable periodic orbits that have different energies. Invariant manifolds, trajectories that asymptotically depart or approach unstable periodic orbits, are used to connect the initial and final orbits. The transfer asymptotically departs the initial orbit on a trajectory contained within the initial orbit’s unstable manifold and later asymptotically approaches the final orbit on a trajectory contained within the stable manifold of the final orbit. The manifold trajectories are connected by the execution of impulsive maneuvers. Two-body parameters dictate the selection of the individual manifold trajectories used to construct efficient transfers. A bounding sphere centered on the secondary, with a radius less than the sphere of influence of the secondary, is used to study the manifold trajectories. A two-body parameter, κ, is computed within the bounding sphere, where the gravitational effects of the secondary dominate. The parameter κ is defined as the sum of two quantities: the difference in the normalized angular momentum vectors and eccentricity vectors between a point on the unstable manifold and a point on the stable manifold. It is numerically demonstrated that as the κ parameter decreases, the total cost to complete the transfer decreases. Preliminary results indicate that this method of constructing transfers produces a significant cost savings over methods that do not employ the use of invariant manifolds.     

An algorithm for solving the pulsar equation

We present an algorithm of finding numerical solutions of pulsar equation. The problem of finding the solutions was reduced to finding expansion coefficients of the source term of the equation in a base of orthogonal functions defined on the unit interval by minimizing a multi-variable mismatch function defined on the light cylinder. We applied the algorithm to Scharlemann and Wagoner boundary conditions by which a smooth solution is reconstructed that by construction passes successfully the Gruzinov’s test of the source function exponent.      

Exceptional Fireball Activity of Orionids in 2006

We report exceptional fireball activity of the Orionid meteor shower in 2006. During four nights in October 2006 the autonomous fireball observatories of the Czech part of the European Fireball Network (EN) recorded 48 fireballs belonging to the Orionids. This is significantly more than the total number of Orionids recorded during about five decades long continuous operation of the EN. Based on precise multi-station photographic and radiometric data we present accurate atmospheric trajectories, heliocentric orbits, light curves and basic physical properties of 10 Orionid fireballs with atmospheric trajectories that were long enough and, with one exception, were observed from at least three stations. Seven were recorded in within a 2-h interval in the night of 20/21 October. Their basic parameters such as radiant positions and heliocentric orbits are very similar. This high fireball activity originated from a very compact geocentric radiant defined by α = 95.10° ± 0.10° and δ = 15.50° ± 0.06°. These fireballs most likely belonged to a distinct filament of larger meteoroids trapped in 1:5 resonance with Jupiter. From detailed light curves and basic fireball classification we found that these meteoroids appertain to the weakest component of interplanetary matter.     

Analysis of theU B V light curves of TT hydrae by Kopal’s frequency domain method

The light curves of the totally eclipsing system TT Hya inUBV colours observed by Kulkarni and Abhyankar during 1973–77 have been analysed by Kopal’s frequency domain method with slight modification. We find y_s (primary) = 0.104 ± 0.005, y_g (secondary) = 0.215 ± 0.008 and i = 89‡ ± 1‡. The value of y_g obtained in this study is smaller than that determined earlier by Kulkarni and Abhyankar by the method of Russell and Merrill; this confirms the undersized nature of the secondary component.The ultraviolet colour excess of the secondary is also confirmed.     

Markov stochastic technique to determine galactic cosmic ray sources distribution

A new numerical model of particle propagation in the Galaxy has been developed, which allows the study of cosmic-ray production and propagation in 2D. The model has been used to solve cosmic ray diffusive transport equation with a complete network of nuclear interactions using the time backward Markov stochastic process by tracing the particles’ trajectories starting from the Solar System back to their sources in the Galaxy. This paper describes a further development of the model to calculate the contribution of various galactic locations to the production of certain cosmic ray nuclei observed at the Solar System.     

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.     

Optimal low-thrust transfers between libration point orbits

Over the past three decades, ballistic and impulsive trajectories between libration point orbits (LPOs) in the Sun–Earth–Moon system have been investigated to a large extent. It is known that coupling invariant manifolds of LPOs of two different circular restricted three-body problems (i.e., the Sun–Earth and the Earth–Moon systems) can lead to significant mass savings in specific transfers, such as from a low Earth orbit to the Moon’s vicinity. Previous investigations on this issue mainly considered the use of impulsive maneuvers along the trajectory. Here we investigate the dynamical effects of replacing impulsive ΔV’s with low-thrust trajectory arcs to connect LPOs using invariant manifold dynamics. Our investigation shows that the use of low-thrust propulsion in a particular phase of the transfer and the adoption of a more realistic Sun–Earth–Moon four-body model can provide better and more propellant-efficient solution. For this purpose, methods have been developed to compute the invariant tori and their manifolds in this dynamical model.     

Algorithms for Stellar Perturbation Computations on Oort Cloud Comets

We investigate different approximate methods of computing the perturbations on the orbits of Oort cloud comets caused by passing stars, by checking them against an accurate numerical integration using Everhart’s RA15 code. The scenario under study is the one relevant for long-term simulations of the cloud’s response to a predefined set of stellar passages. Our sample of stellar encounters simulates those experienced by the Solar System currently, but extrapolated over a time of 10¹⁰ years. We measure the errors of perihelion distance perturbations for high-eccentricity orbits introduced by several estimators – including the classical impulse approximation and Dybczyński’s (1994, Celest. Mech. Dynam. Astron. 58, 1330–1338) method – and we study how they depend on the encounter parameters (approach distance and relative velocity). We introduce a sequential variant of Dybczyński’s approach, cutting the encounter into several steps whereby the heliocentric motion of the comet is taken into account. For the scenario at hand this is found to offer an efficient means to obtain accurate results for practically any domain of the parameter space.     

Stability of a finite disc under the influence of a spherical halo

We have studied the stability of finite gaseous discs, against large-scale perturbations, under the influence of spherical, massive haloes. A surface-density distribution consistent with the observed spiral-tracer profiles in disc galaxies is considered for the disc. We find that growing eigenmodes with both ‘trailing’ and ‘leading’ spirals exist in ‘cold’ discs for a wide range of values of the halo mass and its radius. The amplification rates of the unstable modes reduce as the ratio of the mass of the halo to the mass of the disc is increased. A uniform halo is not very effective towards stabilizing the disc against these modes. The results from the present study are consideredvis-a-vis previous studies on the global modes of self-gravitating discs.     

Multiple solutions in preliminary orbit determination from three observations

Charlier’s theory (1910) provides a geometric interpretation of the occurrence of multiple solutions in Laplace’s method of preliminary orbit determination, assuming geocentric observations. We introduce a generalization of this theory allowing to take into account topocentric observations, that is observations made from the surface of the rotating Earth. The generalized theory works for both Laplace’s and Gauss’ methods. We also provide a geometric definition of a curve that generalizes Charlier’s limiting curve, separating regions with a different number of solutions. The results are generically different from Charlier’s: they may change according to the value of a parameter that depends on the observations.     

Periodic and Chaotic Trajectories of the Second Species for the n-Centre Problem

For the n-centre problem of one particle moving in the potential of attracting centres of small mass fixed in an arbitrary smooth potential and magnetic field, we prove the existence of periodic and chaotic trajectories shadowing sequences of collision orbits. In particular, we obtain large subshifts of solutions of this type for the circular restricted 3-body problem of celestial mechanics. Poincaré had conjectured existence of the periodic ones and given them the name ‘second species solutions’. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Newtonian forces in the Kerr geometry

We study the properties of the ’Newtonian forces’ acting on a test particle in the field of the Kerr black hole geometry. We show that the centrifugal force and the Coriolis force reverse signs at several different locations. We point out the possible relevance of such reversals particularly in the study of the stability properties of the compact rotating stars and the accretion discs in hydrostatic equilibria Received honourable mention in the 1989 Gravity Research Foundation essay competition.     

Polarization-mode separation and the emission geometry of pulsar 0823 + 26: A new pattern of pulsar emission?

We explore the detailed polarization behaviour of pulsar 0823 + 26 using the technique of constructing partial ‘mode-separated’ profiles corresponding to the primary and secondary polarization modes. The characteristics of the two polarization modes in this pulsar are particularly interesting, both because they are anything but orthogonal and because the secondary mode exhibits a structure seen neither in the primary mode nor in the total profile. The new leading and trailing features in the secondary mode, which appear to represent a conal component pair, are interpreted geometrically on the basis of their width and the associated polarization-angle traverse as an outer cone. If the secondary-mode features are, indeed, an outer cone, then questions about the significance of the pulsar’s postcursor component become more pressing. It seems that 0823 + 26 has a very nearly equatorial geometry, in that both magnetic poles and the sightline all fall close to the rotational equator of the star. We thus associate the postcursor component with emission along those bundles of field lines which are also equatorial and which continue to have a tangent in the direction of our sight line for a significant portion of the star’s rotation cycle. It seems that in all pulsars with postcursor components, this emission follows the core component, and all may thus have equatorial emission geometries. No pulsars with precursors in this sense — including the Crab pulsar — are known. The distribution of power between the primary and secondary modes is very similar at both 430 and 1400 MHz. Our analysis shows that in this pulsar considerable depolarization must be occurring on time scales that are short compared to the time resolution of our observations, which is here some 0.5–1.0 milliseconds. One of the most interesting features of the modeseparated partial profiles is a phase offset between the primary and secondary modes. The secondary-mode ‘main pulse’ arrives some 1.5 ± 0.1‡ before the primary-mode one at 430 MHz and some 1.3 +0.1 ‡ at 21 cm. Given that the polar cap has an angular diameter of 3.36‡, we consider whether this is a geometric effect or an effect of differential propagation of the two modes in the inner magnetosphere of the pulsar.