Nonlinear dynamical analysis for displaced orbits above a planet

Nonlinear dynamical analysis and the control problem for a displaced orbit above a planet are discussed. It is indicated that there are two equilibria for the system, one hyperbolic (saddle) and one elliptic (center), except for the degenerate h _z ^max, a saddle-node bifurcation point. Motions near the equilibria for the nonresonance case are investigated by means of the Birkhoff normal form and dynamical system techniques. The Kolmogorov–Arnold–Moser (KAM) torus filled with quasiperiodic trajectories is measured in the τ ₁ and τ ₂ directions, and a rough algorithm for calculating τ ₁ and τ ₂ is proposed. A general iterative algorithm to generate periodic Lyapunov orbits is also presented. Transitions in the neck region are demonstrated, respectively, in the nonresonance, resonance, and degradation cases. One of the important contributions of the paper is to derive necessary and sufficiency conditions for stability of the motion near the equilibria. Another contribution is to demonstrate numerically that the critical KAM torus of nontransition is filled with the (1,1)-homoclinic orbits of the Lyapunov orbit.     

Periodic orbits and bifurcations in the Sitnikov four-body problem

We study the existence, linear stability and bifurcations of what we call the Sitnikov family of straight line periodic orbits in the case of the restricted four-body problem, where the three equal mass primary bodies are rotating on a circle and the fourth (small body) is moving in the direction vertical to the center mass of the other three. In contrast to the restricted three-body Sitnikov problem, where the Sitnikov family has infinitely many stability intervals (hence infinitely many Sitnikov critical orbits), as the “family parameter” ż₀ varies within a finite interval (while z ₀ tends to infinity), in the four-body problem this family has only one stability interval and only twelve 3-dimensional (3D) families of symmetric periodic orbits exist which bifurcate from twelve corresponding critical Sitnikov periodic orbits. We also calculate the evolution of the characteristic curves of these 3D branch-families and determine their stability. More importantly, we study the phase space dynamics in the vicinity of these orbits in two ways: First, we use the SALI index to investigate the extent of bounded motion of the small particle off the z-axis along its interval of stable Sitnikov orbits, and secondly, through suitably chosen Poincaré maps, we chart the motion near one of the 3D families of plane-symmetric periodic orbits. Our study reveals in both cases a fascinating structure of ordered motion surrounded by “sticky” and chaotic orbits as well as orbits which rapidly escape to infinity.     

Relaxation Oscillations in Tidally Evolving Satellites

We study the rotational evolution under tidal torques of axisymmetric natural satellites in inclined, precessing orbits. In the spin- and orbit-averaged equations of motion, we find that a global limit cycle exists for parameter values near the stability limit of Cassini state . The limit cycle involves an alternation between states of near-synchronous spin at low obliquity, and strongly subsynchronous spin at an obliquity near 90°. This dynamical feature is characterized as a relaxation oscillation, arising as the system slowly traverses two saddle-node bifurcations in a reduced system. This slow timescale is controlled by ε, the nondimensional tidal dissipation rate. Unfortunately, a straightforward expansion of the governing equations for small ε is shown to be insufficient for understanding the underlying structure of the system. Rather, the dynamical equations of motion possess a singular term, multiplied by ε, which vanishes in the unperturbed system. We thus provide a demonstration that a dissipatively perturbed conservative system can behave qualitatively differently from the unperturbed system. This revised version was published online in July 2006 with corrections to the Cover Date.     

Chaos in Relativity and Cosmology

Chaos appears in various problems of Relativity and Cosmology. Here we discuss (a) the Mixmaster Universe model, and (b) the motions around two fixed black holes. (a) The Mixmaster equations have a general solution (i.e. a solution depending on 6 arbitrary constants) of Painlevé type, but there is a second general solution which is not Painlevé. Thus the system does not pass the Painlevé test, and cannot be integrable. The Mixmaster model is not ergodic and does not have any periodic orbits. This is due to the fact that the sum of the three variables of the system (α + β + γ) has only one maximum for τ = τ_m and decreases continuously for larger and for smaller τ. The various Kasner periods increase exponentially for large τ. Thus the Lyapunov Characteristic Number (LCN) is zero. The "finite time LCN" is positive for finite τ and tends to zero when τ → ∞. Chaos is introduced mainly near the maximum of (α + β + γ). No appreciable chaos is introduced at the successive Kasner periods, or eras. We conclude that in the Belinskii-Khalatnikov time, τ, the Mixmaster model has the basic characteristics of a chaotic scattering problem. (b) In the case of two fixed black holes M₁ and M₂ the orbits of photons are separated into three types: orbits falling into M₁ (type I), or M₂ (type II), or escaping to infinity (type III). Chaos appears because between any two orbits of different types there are orbits of the third type. This is a typical chaotic scattering problem. The various types of orbits are separated by orbits asymptotic to 3 simple unstable orbits. In the case of particles of nonzero rest mass we have intervals where some periodic orbits are stable. Near such orbits we have order. The transition from order to chaos is made through an infinite sequence of period doubling bifurcations. The bifurcation ratio is the same as in classical conservative systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Secular Dynamics of Asteroids in the Inner Solar System

In the last three years we have carried out numerical and semi-analytical studies on the secular dynamical mechanisms in the region (semimajor axis a < 2 AU) where the NEA orbits evolve. Our numerical integrations (over a time span of a few Myr) have shown that: (i) the linear secular resonances with both the inner and the outer planets may play an important role in the dynamical evolution of NEAs; (ii) the apsidal secular resonance with Mars could provide an important dynamical transport mechanism by which asteroids in the Mars-crossing region eventually achieve Earth-crossing orbits; (iii) in this region, due to the interaction with the terrestrial planets, the Kozai resonance can occur at small inclinations, with the argument of perihelion ω librating around 0° or 180°, providing a temporary protection mechanism against close approaches to the planets. The location of the linear secular resonances in this zone has also been obtained by an automatic procedure using a semi-numerical method valid for all values of the inclinations and eccentricities of the small bodies, and also in the case of libration of the argument of perihelion. A map of the secular resonances in the (a, i) plane shows — in agreement with the numerical integrations — that all the resonances with the terrestrial and giant planets are present, and also that some of them overlap. Thus the way is now open to fully take into account secular resonances in modelling the dynamical evolution of NEAs. This revised version was published online in July 2006 with corrections to the Cover Date.     

The λ0 = 1.35 cm H2O maser line: The hyperfine structure and profile asymmetry

The spectra of several H₂O maser sources exhibit single λ₀ = 1.35 cm maser lines with narrow asymmetric profiles. We consider the hyperfine structure of the line that corresponds to the transition between the rotational 6₁₆ → 5₂₃ levels of ortho-H₂O molecules to account for the line asymmetry. Our numerical simulations of the maser line profile agree well with the observations if the hyperfine structure is taken into account.     

'On Solving Kepler's Equation For Nearly Rectilinear Hyperbolic Orbits'

We deal here with efficient starting points for Kepler's equation in a special case of nearly rectilinear hyperbolic orbits, that is these ones with the eccentricities e1. These orbits appear in stellar dynamics when considering encounters of stars. We test efficiency of these starters for the method for successive approximation (MSA) in its two often applied variants, that is the Newton's method with the quadratic convergence (NM) and in the fixed point method (FPM). Moreover, we determine a dynamical domain of Kepler's equation for this motion.This revised version was published online in October 2005 with corrections to the Cover Date.     

Transition spectra of dynamical systems

The spectra of ‘stretching numbers’ (or ‘local Lyapunov characteristic numbers’) are different in the ordered and in the chaotic domain. We follow the variation of the spectrum as we move from the centre of an island outwards until we reach the chaotic domain. As we move outwards the number of abrupt maxima in the spectrum increases. These maxima correspond to maxima or minima in the curve a(θ), where a is the stretching number, and θ the azimuthal angle. We explain the appearance of new maxima in the spectra of ordered orbits. The orbits just outside the last KAM curve are confined close to this curve for a long time (stickiness time) because of the existence of cantori surrounding the island, but eventually escape to the large chaotic domain further outside. The spectra of sticky orbits resemble those of the ordered orbits just inside the last KAM curve, but later these spectra tend to the invariant spectrum of the chaotic domain. The sticky spectra are invariant during the stickiness time. The stickiness time increases exponentially as we approach an island of stability, but very close to an island the increase is super exponential. The stickiness time varies substantially for nearby orbits; thus we define a probability of escape P_n(x) at time n for every point x. Only the average escape time in a not very small interval Δx around each x is reliable. Then we study the convergence of the spectra to the final, invariant spectrum. We define the number of iterations, N, needed to approach the final spectrum within a given accuracy. In the regular domain N is small, while in the chaotic domain it is large. In some ordered cases the convergence is anomalously slow. In these cases the maximum value of a_k in the continued fraction expansion of the rotation number a = [a₀,a₁,... a_k,...] is large. The ordered domain contains small higher order chaotic domains and higher order islands. These can be located by calculating orbits starting at various points along a line parallel to the q-axis. A monotonic variation of the sup {q}as a function of the initial condition q₀ indicates ordered motions, a jump indicates the crossing of a localized chaotic domain, and a V-shaped structure indicates the crossing of an island. But sometimes the V-shaped structure disappears if the orbit is calculated over longer times. This is due to a near resonance of the rotation number, that is not followed by stable islands. This revised version was published online in July 2006 with corrections to the Cover Date.     

Long-term dynamical evolution of dusty ejecta from Deimos

We re-assess expected properties of the presumed dust belt of Mars formed by impact ejecta from Deimos. Previous studies have shown that dynamics of Deimos particles are dominated by two perturbing forces: radiation pressure (RP) and Mars’ oblateness (J2). At the same time, they have demonstrated that lifetimes of particles, especially of grains about ten of micrometers in size, may reach more than 10⁴ years. On such timescales, the Poynting-Robertson drag (PR) becomes important. Here we provide a study of the dynamics under the combined action of all three perturbing forces. We show that a PR decay of the semimajor axes leads to an adiabatic decrease of amplitudes and periods of oscillations in orbital inclinations predicted in the framework of the underlying RP+J2 problem. Furthermore, we show that smallest of the long-lived Deimos grains (radius≈5- may reach a chaotic regime, resulting in unpredictable and abrupt changes of their dynamics. The particles just above that size (≈10-) should be the most abundant in the Deimos torus. Our dynamical analysis, combined with a more accurate study of the particle lifetimes, provides corrections to earlier predictions about the dimensions and geometry of the Deimos torus. In addition to a population, appreciably inclined and shifted towards the Sun, the torus should contain a more contracted, less asymmetric, and less tilted component between the orbits of Phobos and Deimos.     

Coulomb-trapped particles in the electromagnetic fields of an oblique magnetic rotator

An oblique, rotating magnetized sphere emits electromagnetic waves which, for large magnetization, can quickly accelerate charged particles to very high energies. A central, attractive Coulomb force can trap particles in the region beyond the light cylinder by balancing the accelerating influence of the radiation on the particles. We sample some of the particle orbits possible under these dynamical conditions. A general feature of these orbits is that non-interacting particles started with random initial conditions in the domain of attraction of these orbits will arrange themselves on a curve corotating with the axis of magnetization. Such particle configurations can be a source of pulsed radiation. In the idealized case of no interparticle interactions the spectral index for the radiation emitted by one frequently occurring configuration is found to be –2/3, for emission from radio to -ray frequencies. The dynamical conditions in this simple model closely match those prevalent in outer pulsar magnetospheres, making it possible that part of the radiation from pulsars is emitted by trapped plasma in the region beyond the light cylinder.     

Molecular Gas in the Merging System ARP 299

We present new observations of the molecular gas distribution in the merging system Arp 299. The first observation set was obtained with the Canada–France–Hawaii Telescope near-IR camera Redeye and the second set comes from the IRAM Plateau de Bure interferometer (combined with short spacings observed at the IRAM 30 m Telescope). In the near IR, H₂ ν=1→0 S(1) and Br_γ line maps are globally identical: there is bright emission not only at the two galaxy nuclei but also in numerous extranuclear star forming regions. Moreover, there is weaker emission localized in filaments between and around the two nuclei. These filaments correspond to a dust lane observed in optical images from HST. ¹²CO(1→0), ¹³CO(1→0) and HCN(1→0) maps are also presented. The structure of the¹²CO(1→0) map is very close to the NIR observations: the same bright galaxy nuclei and star-forming regions, the same filaments, but half of the total flux is found in weak extended emission. Strong HCN emission is observed in the nucleus A indicating the presence of a large amount of dense gas. Nucleus B1 is weak in ¹²CO(1→0) emission while nucleus A and star-forming regions C-C′ show more normal ¹³CO/¹²CO ratios. This revised version was published online in July 2006 with corrections to the Cover Date.     

Families of Asymmetric Periodic Orbits in Hill’s Problem of Three Bodies

We present five families of periodic solutions of Hill’s problem which are asymmetric with respect to the horizontal ξ axis. In one of these families, the orbits are symmetric with respect to the vertical η axis; in the four others, the orbits are without any symmetry. Each family consists of two branches, which are mirror images of each other with respect to the ξ axis. These two branches are joined at a maximum of Γ, where the family of asymmetric periodic solutions intersects a family of symmetric (with respect to the ξ axis) periodic solutions. Both branches can be continued into second species families for Γ → − ∞.     

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.     

Physical properties and orbital stability of the Trojan asteroids

All the Trojan asteroids orbit about the Sun at roughly the same heliocentric distance as Jupiter. Differences in the observed visible reflection spectra range from neutral to red, with no ultra-red objects found so far. Given that the Trojan asteroids are collisionally evolved, a certain degree of variability is expected. Additionally, cosmic radiation and sublimation are important factors in modifying icy surfaces even at those large heliocentric distances. We search for correlations between physical and dynamical properties, we explore relationships between the following four quantities; the normalised visible reflectivity indexes (S^′), the absolute magnitudes, the observed albedos and the orbital stability of the Trojans. We present here visible spectroscopic spectra of 25 Trojans. This new data increase by a factor of about 5 the size of the sample of visible spectra of Jupiter Trojans on unstable orbits. The observations were carried out at the ESO-NTT telescope (3.5 m) at La Silla, Chile, the ING-WHT (4.2 m) and NOT (2.5 m) at Roque de los Muchachos observatory, La Palma, Spain. We have found a correlation between the size distribution and the orbital stability. The absolute-magnitude distribution of the Trojans in stable orbits is found to be bimodal, while the one of the unstable orbits is unimodal, with a slope similar to that of the small stable Trojans. This supports the hypothesis that the unstable objects are mainly byproducts of physical collisions. The values of S^′ of both the stable and the unstable Trojans are uniformly distributed over a wide range, from 0%/1000 Å to about 15%/1000 Å. The values for the stable Trojans tend to be slightly redder than the unstable ones, but no significant statistical difference is found.     

A New Pumping Mechanism for A Series of Class II Methanol Masers in the J 0 - J -1 E Transitions

A new pumping mechanism – methanol masers without population inversion is presented in this paper. It can be used to explain the formation of a series of J ₀ → J _-1 E methanol masers, while the 2₁ → 3₀ A ⁺ methanol masers are regarded as a driving coherent micrwave field. In the new mechanism, the intensities of J ₀ - J _-1 E methanol masers are increased with the decreasing transition frequencies (or with rotational number J, approximately). These results agree with Slysh et al. (1995) and Slysh et al. (1999) J ≤ 5 observations for G3345.01+1.79 and W48, in which both J ₀ → J _-1 E and 2₁ → 3₀ A ⁺ methanol masers are detected coincidentally. Other astronomical conditions, such as magnetic field, 2₁ → 3₀ A ⁺ coherent radition, incoherent pumping rate by thermal radition and so on are also discussed. The new mechanism can operate as a complement to other ordinary maser pumping mechanisms for some class II methonal maser sources. This revised version was published online in July 2006 with corrections to the Cover Date.     

Periodic orbits in a resonant dynamical system

We study the periodic orbits in a two dimensional dynamical system, symmetric with respect to both axes, with two equal or nearly equal frequencies. It is shown that the periodic orbits can be found directly from the equations of motion. The form of these orbits depends on the value of the coupling parameter . We verify the theoretical results by numerical calculations.     

Motion Around The Triangular Equilibrium Points Of The Restricted Three-Body Problem Under Angular Velocity Variation

We study numerically the asymmetric periodic orbits which emanate from the triangular equilibrium points of the restricted three-body problem under the assumption that the angular velocity ω varies and for the Sun–Jupiter mass distribution. The symmetric periodic orbits emanating from the collinear Lagrangian point L ₃, which are related to them, are also examined. The analytic determination of the initial conditions of the long- and short-period Trojan families around the equilibrium points, is given. The corresponding families were examined, for a combination of the mass ratio and the angular velocity (case of equal eigenfrequencies), and also for the critical value ω = 2

The Apsidal Antialignment of the HD 82943 System

We perform numerical simulations to explore the dynamical evolution of the HD 82943 planetary system. By simulating diverse planetary configurations, we find two mechanisms of stabilizing the system: the 2:1 mean motion resonance (MMR) between the two planets can act as the first mechanism for all stable orbits. The second mechanism is a dynamical antialignment of the apsidal lines of the orbiting planets, which implies that the difference of the periastron longitudes ₃ librates about 180° in the simulations. We also use a semi-analytical model to explain the numerical results for the system under study.     

Newtonian and relativistic periodic orbits around two fixed black holes

We compare families of simple periodic orbits of test particles in the Newtonian and relativistic problems of two fixed centers (black holes). The Newtonian problem is integrable, while the relativistic problem is highly non-integrable.The orbits are calculated on the meridian plane through the fixed centersM ₁ (atz=+1) andM ₂ (atz=–1) for energies smaller than the escape energyE=1. We use prolate spheroidal coordinates (, , =const) and also the variables =cosh and =–cos . The orbits are inside a curve of zero velocity (CZV). The Newtonian orbits are also limited by an ellipse and a hyperbola, or by two eillipses. There are 3 main types of periodic orbits (1) elliptic type (around both centers), (2) hyperbolic-type, and (3) resonant-type.The elliptic type orbits are stable in the Newtonian case and both stable and unstable in the relativistic case. From the stable orbits bifurcate double period orbits both symmetric and asymmetric with respect to thez-axis. There are also higher order bifurcations. The hyperbolic-type orbits are unstable. The Newtonian resonant orbits are defined by the ratiot _µ/t =n/m of oscillations along and during one period, and they are all marginally unstable. The corresponding relativistic orbits are stable, or unstable. The main families are figure eight orbits aroundM ₁, or aroundM ₂ (3/1 orbits); gamma, or inverse gamma orbits (4/2); higher resonant families 5/1,7/1,...,8/2,12/2,...;, more complicated orbits, like 5/3, and bifurcations from the above orbits. Satellite orbits aroundM ₁, orM ₂, and their bifurcations (e.g. double period) exist in the relativistic case but not in the Newtonian case. The characteristics of the various families are quite different in the Newtonian and the relativistic cases. The sizes of the orbits and their stabilities are also quite different in general. In the Appendix we study the various types of straight line orbits and prove that some subcases introduced by Charlier (1902) are impossible.