Orbital structure in oscillating galactic potentials

Many of exoplanetary systems consist of more than one planet and the study of planetary orbits with respect to their long-term stability is very interesting. Furthermore, many exoplanets seem to be locked in a mean-motion resonance (MMR), which offers a phase protection mechanism, so that, even highly eccentric planets can avoid close encounters. However, the present estimation of their initial conditions, which may change significantly after obtaining additional observational data in the future, locate most of the systems in chaotic regions and consequently, they are destabilized. Hence, dynamical analysis is imperative for the derivation of proper planetary orbital elements. We utilize the model of spatial general three body problem, in order to simulate such resonant systems through the computation of families periodic orbits. In this way, we can figure out regions in phase space, where the planets in resonances should be ideally hosted in favour of long-term stability and therefore, survival. In this review, we summarize our methodology and showcase the fact that stable resonant planetary systems evolve being exactly centered at stable periodic orbits. We apply this process to co-orbital motion and systems HD 82943, HD 73526, HD 128311, HD 60532, HD 45364 and HD 108874.

The Poincaré recurrence statistic (PRS) measures the probability that a trajectory initiated in a phase space region will first return to that region, as a function of time. For deterministic, area-preserving maps with a mixture of regular and chaotic orbits, the stickiness of invariant tori and islands is responsible for a power-law decay in the PRS. We show that noise perturbations allow trajectories to access the interior of islands, enhancing their trapping effect and causing many orbits to take longer to return to a neighborhood of their initial conditions. The noisy PRS can then exhibit an extended tail on an intermediate time scale, which is eventually followed by an asymptotic exponential decay. We study a typical example, the Harper map, and compare distributions of trapping and visit times to islands with recurrence times to show the importance of noise in creating tails in the PRS. A simple finite-state Markov model of the dynamics confirms how this slower decay can be caused by noise permitting entry to previously inaccessible regions.

We apply the Mean Exponential Growth Factor of Nearby Orbits (MEGNO) technique to the dynamics of Jovian irregular satellites. The MEGNO indicator is a practical numerical tool to distinguish between quasi-periodic and chaotic structures in phase space of a given dynamical system. The MEGNO indicator is used to generate a mapping of relevant phase-space regions occupied by observed Jovian irregular satellites. The construction of MEGNO maps of the Jovian phase-space region within its Hill-sphere is addressed and the obtained results are compared with previous studies regarding the dynamical stability of irregular satellites. Since this is the first time the MEGNO technique is applied to study the dynamics of irregular satellites, we provide a review of the MEGNO theory and illustrate basic properties. We consider the elliptic restricted three-body problem in which Jupiter is orbited by a massless test satellite subject to solar gravitational perturbations. The equations of motion of the system are integrated numerically and the MEGNO indicator computed from the system's variational equations. A large set of initial conditions is studied to generate the MEGNO maps. The chaotic nature of initial conditions is demonstrated by studying a quasi-periodic orbit and a chaotic orbit. As a result, we establish the existence of several high-order mean-motion resonances (MMR) detected for retrograde orbits along with other interesting dynamical features related to various dynamical resonances. The computed MEGNO maps allow us to differentiate qualitatively between chaotic and quasi-periodic regions of the irregular satellite phase space within a relatively short integration time of 60 000 yr for each orbit. By comparing with previous published results, we can establish a correlation between chaotic regions and corresponding regions of orbital instability. Based on our results, we hypothesize on the possibility of gravitational scattering from high-order MMR as a dynamical cause to explain the observed orbital velocity dispersion for members of the Pasiphae family.

Symmetry in regular motion; Relation to chaos

We created a triaxial stellar system through the cold dissipationless collapse of 100,000 particles whose evolution was followed with a multipolar code. Once an equilibrium system had been obtained, the multipolar expansion was freezed and smoothed in order to get a stationary smooth potential. The resulting model was self-consistent and the orbits and Lyapunov exponents could then be computed for a randomly selected sample of 3472 of the bodies that make up the system. More than half of the orbits (52.7 % ) turned out to be chaotic. Regular orbits were then classified using the frequency analysis automatic code of Carpintero and Aguilar (1998, MNRAS 298(1), 1–21). We present plots of the distributions of the different kinds of orbits projected on the symmetry planes of the system. We distinguish chaotic orbits with only one non-zero Lyapunov exponent from those with two non-zero exponents and show that their spatial distributions differ, that of the former being more similar to the one of the regular orbits. Most of the regular orbits are boxes and boxlets, but the minor axis tubes play an important role filling in the wasp waists of the boxes and helping to give a lentil shape to the system. We see no problem in building stable triaxial models with substantial amounts of chaotic orbits; the difficulties found by other authors may be due not to a physical cause but to a limitation of Schwarzschild’s method.

We analyze the orbits of stars and dark matter particles in the halo of a\ndisk galaxy formed in a cosmological hydrodynamical simulation. The halo is\noblate within the inner ~20 kpc and triaxial beyond this radius. About 43% of\norbits are short axis tubes - the rest belong to orbit families that\ncharacterize triaxial potentials (boxes, long-axis tubes and chaotic orbits),\nbut their shapes are close to axisymmetric. We find no evidence that the\nself-consistent distribution function of the nearly oblate inner halo is\ncomprised primarily of axisymmetric short-axis tube orbits. Orbits of all\nfamilies, and both types of particles are highly eccentric with mean\neccentricity >0.6. We find that randomly selected samples of halo stars show no\nsubstructure in "integrals of motion" space. However individual accretion\nevents can be clearly identified in plots of metallicity versus formation time.\nDynamically young tidal debris is found primarily on a single type of orbit.\nHowever, stars associated with older satellites become chaotically mixed during\nthe formation process (possibly due to scattering by the central bulge and\ndisk, and baryonic processes), and appear on all four types of orbits. We find\nthat the tidal debris in cosmological hydrodynamical simulations experiences\nsignificantly more chaotic evolution than in collisionless simulations, making\nit much harder to identify individual progenitors using phase space coordinates\nalone. However by combining information on stellar ages and chemical abundances\nwith the orbital properties of halo stars in the underlying self-consistent\npotential, the identification of progenitors is likely to be possible.\n

A dynamical mechanism is proposed that explains the spiral structure observed frequently as a continuation of the bars in barred spiral galaxies. It is argued that the part of the spirals attached to the bar is due to chaotic orbits. These are chaotic orbits that exhibit for long time intervals a 4:1-resonance orbital behaviour. They are of the same type of orbit as is responsible for the boxiness of the outer isophotes of the bar in cases like NGC 4314, as indicated by Patsis, Athanassoula & Quillen. The spirals formed this way are faint with respect to the bar, open as they wind out, and do not extend over an angle larger than π/2. A possible continuation of the spiral structure towards larger angles can be due to orbits trapped around stable periodic orbits at the corotation region. We present a family of stable, banana-like periodic orbits, precessing as EJ increases, that can play this role.

We study the orbits of photons (null geodesics) and particles (time-like geodesics) in the field of two fixed black holes. The orbits of photons terminate either at the black holes M 1 and M 2 (types (I) and (II)), or at infinity (type (III)). The limits of these three types of orbits are Cantor sets, defined by the unstable periodic orbits, that form a set of measure zero. In the case of particles with elliptic energy there are some stable periodic orbits that trap around them both quasiperiodic orbits (type (IV)) and chaotic orbits that do not reach the black holes (type (V)). The stable periodic orbits may become unstable, producing an infinity of period doubling bifurcations. The main periodic orbits are two almost circular satellite orbits around each black hole, M 1 and M 2, and almost elliptical orbits surrounding both black holes. Orbits crossing one of the satellite orbits inwards fall into the corresponding black hole. For certain values of the parameters one or both satellite orbits may not exist. Then the separation of the type (I) and/or type (II) orbits is done by some almost hyperbolic orbits. In the Newtonian case the satellite orbits around M 1 and M 2 never exist (except if M 1 = 0, or M 2 = 0 exactly). However we have many families of higher order periodic orbits that exist both in the relativistic and the Newtonian case. The differences between the corresponding orbits are large close to the black holes.

This paper continues an investigation of chaos and chaotic phase mixing in triaxial generalisations of the Dehnen potential which have been proposed to describe realistic ellipticals that have a strong density cusp and manifest significant deviations from axisymmetry. Earlier work is extended in three important ways, namely by exploring systematically the effects of (1) variable axis ratios, (2) ‘graininess ’ associated, e.g., with stars and bound substructures, idealised as friction and white noise, and (3) large-scale organised motions within a galaxy and a dense cluster environment, each presumed to induce near-random forces idealised as coloured noise with a finite autocorrelation time. The effects of varying the axis ratio were studied in detail by considering two sequences of models with cusp exponent γ = 1 and, respectively, axis

The disturbance influence on vibration of a belt device driven by a crank mechanism

The biomass replaces the petroleum as energy is involved in energy transformations. There are two grade transformations from the wheat which are the fuel ethanol and the marsh gas ferment. The study on their transformation mechanisms and how to improve their energy efficiency will be benefit for both the academic and practical significance. Studies have shown that at least 67.05% wheat energy changes into fuel ethanol and at least 15.46% the rest of accessory substance alcohol lees energy changes into marsh gas. The total energy transformation amount is not less than 82.51%.The ratio of the wheat fuel ethanol input to output is 1.04, and NEV is 1167 MJ. The result shows the values of energy gain are positive.

Despite a lack of traditional speech features, novel sentences restricted to a narrow spectral slit can retain nearly perfect intelligibility [R. M. Warren et al., Percept. Psychophys. 57, 175-182 (1995)]. The current study employed 514 listeners to elucidate the cues allowing this high intelligibility, and to examine generally the use of narrow-band temporal speech patterns. When 1/3-octave sentences were processed to preserve the overall temporal pattern of amplitude fluctuation, but eliminate contrasting amplitude patterns within the band, sentence intelligibility dropped from values near 100% to values near zero (experiment 1). However, when a 1/3-octave speech band was partitioned to create a contrasting pair of independently amplitude-modulated 1/6-octave patterns, some intelligibility was restored (experiment 2). An additional experiment (3) showed that temporal patterns can also be integrated across wide frequency separations, or across the two ears. Despite the linguistic content of single temporal patterns, open-set intelligibility does not occur. Instead, a contrast between at least two temporal patterns is required for the comprehension of novel sentences and their component words. These contrasting patterns can reside together within a narrow range of frequencies, or they can be integrated across frequencies or ears. This view of speech perception, in which across-frequency changes in energy are seen as systematic changes in the temporal fluctuation patterns at two or more fixed loci, is more in line with the physiological encoding of complex signals.

The two-bodies problem can be fully solved, and was solved by Kepler (1609) and Newton (1687). The general three-body problem is often given as an example of a mathematical problem that ‘can’t be solved’. So, there is no general analytical solution. This problem can be significant and a special case of this problem is the Circular Restricted Three-Body Problem (CRTBP), which can be applied to the Earth-Moon system with a spacecraft, the Sun-Earth system with an asteroid, etc. In this paper, let’s focus on the motion of a test particle near the triangular Lagrange points L4 and L5 in the Earth-Moon and the Sun-Earth systems. Studying the movement of an object around these points is especially important for space mission design. To generate a trajectory around these points, the non-linear equations of motion for the circular restricted three-body problem were numerically integrated into MATLAB® 2023 software and the results are presented in the plane (x, y) and the phase plane (x, vx) and (y, vy). By numerical orbit integration, it is possible to investigate what happens when the displacement is relatively large or short from the Lagrange points. Then the small astronomical body may vibrate around these points. The results in this paper are shown in the rotating and inertia axes. Various initial positions near the Lagrange points and velocities are used to produce various paths the test particle can take. The same examples of numerical studies of trajectories associated with Lagrange points are shown in the inertial and the rotating coordinates system and are discussed. From the results of the numerical tests performed in MATLAB® 2023, it is possible to saw that there are different types of periodic, quasi-periodic, and chaotic orbits

Extensively optimized Lewis orbital (Frost model) structures are reported for CH3N, CH3NH+, CH3CH, CH3CH, CH3BH2, CH2NH, and CH2NH (spiro and planar). Electronic energy differences between these isoelectronic species were estimated by the integral Hellmann‐Feynman (iHF) formula, with the hope that satisfaction of the Hellmann‐Feynman conditions would lead to accurate iHF values of energy changes. We observed a strongly nonlinear relation between the iHF error and the departure of the overlap of wave‐functions of the structures from unity. MO computations in common orbital (not determinantal) basis sets for CH3NH+CH2NH (planar), CH3NCH2NH, and CH3CHCH2CH2 produced greatly improved iHF estimates of energy changes, reducing errors by as much as 80 times. Certain features of the static optimum structures and the transition densities suggested that the syn path for rearrangement of methyl carbene to ethylene is a general feature of rearrangements in these systems.

Understanding the regular or chaotic nature of orbits in galaxies is undoubtedly an issue of great importance. We determine the character of orbits of stars moving in the meridional plane (R, z) of an axially symmetric time-independent galactic model with a spherical central nucleus, and a flat biaxial oblate dark matter halo component. In particular, we try to reveal the influence of the fractional portion of dark matter on the structure and also on the different families of orbits of the galaxy, by monitoring how the percentage of chaotic orbits, as well as the percentages of orbits of the main regular resonant families evolve when the ratio of dark matter to luminous mass varies. The smaller alignment index (SALI) was computed by numerically integrating the equations of motion as well as the variational equations to extensive samples of orbits in order to distinguish between ordered and chaotic motion. In addition, a method based on the concept of spectral dynamics that utilizes the Fourier transform of the time series of each coordinate is used to identify the various families of regular orbits and also to recognize the secondary resonances that bifurcate from them. The investigation is carried out both in the physical (R, z) and the phase \((R,\dot {R})\) space for a better understanding of the orbital properties of the system. The numerical computations reveal that in both cases, the fractional portion of dark matter influences more or less, the overall orbital structure of the system. It was observed however, that the evolution of the percentages of all types of orbits as a function of the fractional portion of dark matter strongly depends on the particular type of space (physical or phase) in which the initial conditions of orbits are launched. The results are compared with the similar earlier work.