Nearby Orbits Research Articles

Studying Binary Formation under Dynamical Friction Using Hill’s Problem

Using the equations of motion from Hill’s problem, with added accelerations for different forms of dynamical friction, we provide the (to-date) broadest scale-free study of friction-driven binary formation in gaseous disks and stellar clusters. We focus mainly on binary formation between stellar-mass black holes in active galactic nuclei (AGNs), considering both gas dynamical friction (GDF) from AGN disks and stellar dynamical friction (SDF) from the nuclear star cluster. We first find simple, dimensionless friction coefficients that approximate the effects of standard models for GDF and SDF. We perform extensive simulations of Hill’s problem under such friction, and we present a picture of binary formation through encounters between single stars on nearby orbits, as a function of friction parameter, eccentricity, and inclination. Notably, we find that the local binary formation rate is a linear function of the friction coefficient so long as the friction is weak. Due to the dimensionless nature of our model problem, our findings are generalizable to binary formation at all scales (e.g., intermediate-mass black holes in a star cluster, planetesimals in a gaseous disk).

Dynamical coupling of Keplerian orbits in a hierarchical four-body system: from the Galactic Centre to compact planetary systems

ABSTRACT This study focuses on the long-term evolution of two bodies in nearby initially coplanar orbits around a central dominant body perturbed by a fourth body on a distant Keplerian orbit. Our previous works that considered this setup enforced circular orbits by adding a spherical potential of extended mass, which dampens Kozai–Lidov oscillations; it led to two qualitatively different modes of the evolution of the nearby orbits. In one scenario, their mutual interaction exceeds the effect of differential precession caused by a perturbing body. This results in a long-term coherent evolution, with nearly coplanar orbits experiencing only small oscillations of inclination. We extend the previous work by (i) considering post-Newtonian corrections to the gravity of the central body, either instead of or in addition to the potential of extended mass, (ii) relaxing the requirement of strictly circular orbits, and (iii) removing the strict requirement of complete Kozai–Lidov damping. Thus, we identify the modes of interorbital interaction described for the zero eccentricity case in the more general situation, which allows for its applicability to a much broader range of astrophysical systems than considered initially. In this work, we scale the systems to the orbits of S-stars; we consider the clockwise disc to represent the perturbing body, with post-Newtonian corrections to the gravity of Sagittarius A* playing the role of damping potential. Considering post-Newtonian corrections, even stellar-mass central bodies in compact planetary systems can allow for the coupled evolution of Keplerian orbits.

Occurrence Modes of AAEMs (Na+ and Ca2+) and the Effect on the Molecular Structures of Zhundong Coal via Quantum Chemistry.

Zhundong coal is known for its high content of alkali and alkaline earth metals (AAEMs), which greatly influences coal processing and utilization. To reveal the occurrence modes and the effect of AAEM ions on the molecular structures of Zhundong coal, the previously constructed molecular structure models of vitrinite-rich and inertinite-rich Zhundong coal (ZD-V and ZD-I) were selected to simulate using quantum chemical methods. By focusing on Na+ and Ca2+, the adsorption capacity at different adsorption sites was investigated based on the density functional theory (DFT), and the effects of adsorption of Na+ and Ca2+ on nearby atomic charges, chemical bonds, and molecular orbitals were investigated. Results show that compared with ZD-I, ZD-V contains a more negative electrostatic potential (ESP) distribution and lower bond order, indicating that vitrinite contains more adsorption sites for AAEM ions and exhibits stronger chemical reactivity. Na+ and Ca2+ are easily adsorbed to the most negative ESP with the optimal adsorption site near the carbonyl group (C=O). Compared with adsorbed Na+, Ca2+ has a smaller adsorption distance from the molecule and a higher adsorption energy. Ca2+ can transfer more charge than Na+ and has more affinity with the coal molecule. Ca2+ at all adsorption sites is bound to organic molecules by chemisorption, which also reveals the reason for the low water-soluble Ca content in coal at the molecular level. Adsorption of AAEM ions has a more significant effect on the chemical bond of oxygen-containing functional groups near AAEM ions compared to the overall molecular fragments, which makes the nearby chemical bonds (C-O/C=O) decrease in bond order and increase in bond length. Ca2+ makes the nearby chemical bonds more prone to break than Na+. Additionally, Ca2+ has a more significant impact on the energy gaps (ΔEgap) compared to Na+. Adsorption of Ca2+ near the carbonyl and carboxyl groups leads to a significant decrease in ΔEgap, indicating an enhanced chemical reactivity of coal molecular fragments.

Oscillations in three-reaction quadratic mass-action systems.

It is known that rank-two bimolecular mass-action systems do not admit limit cycles. With a view to understanding which small mass-action systems admit oscillation, in this paper we study rank-two networks with bimolecular source complexes but allow target complexes with higher molecularities. As our goal is to find oscillatory networks of minimal size, we focus on networks with three reactions, the minimum number that is required for oscillation. However, some of our intermediate results are valid in greatergenerality. One key finding is that an isolated periodic orbit cannot occur in a three-reaction, trimolecular, mass-action system with bimolecular sources. In fact, we characterize all networks in this class that admit a periodic orbit; in every case, all nearby orbits are periodic too. Apart from the well-known Lotka and Ivanova reactions, we identify another network in this class that admits a center. This new network exhibits a vertical Andronov-Hopfbifurcation. Furthermore, we characterize all two-species, three-reaction, bimolecular-sourced networks that admit an Andronov-Hopf bifurcation with mass-action kinetics. These include two families of networks that admit a supercritical Andronov-Hopf bifurcation and hence a stable limit cycle. These networks necessarily have a target complex with a molecularity of at least four, and it turns out that there are exactly four such networks that aretetramolecular.

Performance of chaos diagnostics based on Lagrangian descriptors. Application to the 4D standard map

We investigate the ability of simple diagnostics based on Lagrangian descriptor (LD) computations of initially nearby orbits to detect chaos in conservative dynamical systems with phase space dimensionality higher than two. In particular, we consider the recently introduced methods of the difference and the ratio of the LDs of neighboring orbits, as well as a quantity related to the finite-difference second spatial derivative of the LDs, and use them to determine the chaotic or regular nature of ensembles of orbits of a prototypical area-preserving map model, the 4-dimensional symplectic standard map. We compare the characterization obtained by these LDs-based diagnostics against that achieved by the Smaller Alignment Index method of chaos detection, by recording the percentage agreement PA between the two classifications. We also study the influence of the final number of orbit iterations T, the order n of the indices, as well as the distance σ of neighboring orbits on the performance of these methods, and find appropriate T, n and σ values which allow the efficient use of the three indices as short time and computationally cheap chaos diagnostics achieving PA≳90%. Our findings clearly indicate the capability of LDs to efficiently identify chaos in systems whose phase space is difficult to visualize, due to its high dimensionality, without knowing the variational equations (tangent map) of continuous (discrete) time systems needed by traditional chaos indicators.

Influence of planets on debris discs in star clusters – I. The 50 au Jupiter

ABSTRACT Although debris discs may be common in exoplanet systems, only a few systems are known in which debris discs and planets coexist. Planets and the surrounding stellar population can have a significant impact on debris disc evolution. Here, we study the dynamical evolution of debris structures around stars embedded in star clusters, aiming to determine how the presence of a planet affects the evolution of such structures. We combine NBODY6++GPU and REBOUND to carry out N-body simulations of planetary systems in star clusters ($N=8\, 000$; Rh = 0.78 pc) for a period of 100 Myr, in which 100 solar-type stars are assigned 200 test particles. Simulations are carried out with and without a Jupiter-mass planet at 50 au. We find that the planet destabilizes test particles and speeds up their evolution. The planet expels most particles in nearby and resonant orbits. Remaining test particles tend to retain small inclinations when the planet is present, and fewer test particles obtain retrograde orbits. Most escaping test particles with speeds smaller than the star cluster’s escape speed originate from cold regions of the planetary system or from regions near the planet. We identify three regions within planetary systems in star clusters: (i) the private region of the planet, where few debris particles remain (40–60 au), (ii) the reach of the planet, in which particles are affected by the planet (0–400 au), and (iii) the territory of the planetary system, most particles outside which will eventually escape (0–700 au).

Chaos in multiplanetary extrasolar systems

ABSTRACT Here, we present an initial look at the dynamics and stability of 178 multiplanetary systems that are already confirmed and listed in the NASA Exoplanet Archive. To distinguish between the chaotic and regular nature of a system, the value of the Mean Exponential Growth factor of Nearby Orbits (MEGNO) indicator for each system was determined. Almost three-quarters of them could be labelled as long-term stable. Only 45 studied systems show chaotic behaviour. We consequently investigated the effects of the number of planets and their parameters on the system stability. A comparison of results obtained using the MEGNO indicator and machine-learning algorithm SPOCK (Stability of Planetary Orbital Configuration Klassifier) suggests that the SPOCK could be used as an effective tool for reviewing the stability of multiplanetary systems. A similar study was already published by Laskar and Petit. We compared their analysis based on the AMD (Angular Momentum Deficit) criterion with our results. The possible discrepancies are discussed.

Isomorphisms on interpolation spaces generated by the method of means

We investigate the stability of isomorphisms acting between interpolation spaces generated by the method of means. We focus on the methods which are determined by balanced sequences for non-degenerate quasi-concave functions. The key point for our investigation is that these methods have orbital description by a single element generated by a special ideal of operators between Banach couples. We prove that if an operator is invertible in one orbit it is also invertible by nearby orbits provided that the corresponding indices of quasi-concave functions generated these orbits are close to each other. In particular, these results apply to the real method of interpolation.

Asymptotic approximations for the phase space Schrödinger equation

We consider semi-classical time evolution for the phase space Schrödinger equation and present two methods of constructing short time asymptotic solutions. The first method consists of constructing a semi-classical phase space propagator in terms of semi-classical Gaussian wave packets on the basis of the anisotropic Gaussian approximation, related to the nearby orbit approximation, which is identified with the phase space representation of the Littlejohn semi-classical propagator. By means of the semi-classical propagator, we derive an asymptotic solution for configuration space WKB initial data. The second method consists of constructing a phase space narrow beam asymptotic solution, following the complex WKB theory developed by Maslov, on the basis of a canonical system in double phase space, analogous to the Berezin–Shubin–Marinov Hamilton–Jacobi and transport equations. We illustrate the methods for sub-quadratic potentials in .

Discrete-Variable-Thrust Guidance for Orbital Rendezvous Based on Feedback Linearization

This research is focused on the analysis, design, and numerical testing of a feedback guidance algorithm for autonomous (unmanned) close-range maneuvering of a chaser spacecraft, in the context of orbital rendezvous with a target vehicle. The relative dynamics of the two vehicles, placed in nearby low Earth orbits, is modeled using the nonlinear Battin–Giorgi equations of relative motion, with the inclusion of all the relevant perturbations, i.e. several harmonics of the geopotential, atmospheric drag, solar radiation pressure, and third body gravitational pull due to Moon and Sun. Unlike several former contributions in the scientific literature, this research considers the orbit perturbing actions on both vehicles, proving that this is crucial for a successful maneuver. Feedback linearization provides the theoretical foundation for the definition and development of a guidance algorithm that is capable of driving the chaser vehicle toward the target spacecraft. Moreover, discrete-variable thrust is considered, and an effective modulation scheme is proposed that includes adaptation of the control gains. Monte Carlo simulations demonstrate that the guidance technique at hand is effective and accurate in driving the chaser spacecraft toward the target vehicle, in the presence of orbit perturbations and unpredictable displacements from the nominal initial conditions.

A numerical study of the 1/2, 2/1, and 1/1 retrograde mean motion resonances in planetary systems

ABSTRACT We present a numerical study on the stability of the 1/2, 2/1, and 1/1 retrograde mean motion resonances in the three-body problem composed of a solar mass star, a Jupiter mass planet, and an additional body with zero mass (elliptic restricted three-body problem) or masses corresponding to either Neptune, Saturn, or Jupiter (planetary three-body problem). For each system, we obtain stability maps using the n-body numerical integrator REBOUND and computing the chaos indicator mean exponential growth factor of nearby orbits (MEGNO). We show that families of periodic orbits exist in all configurations and they correspond to the libration of either a single resonant argument or all resonant arguments (fixed points). We compare the results obtained in the elliptic restricted three-body problem with previous results in the literature, and we show the differences and similarities between the phase space topology for these retrograde resonances in the circular restricted, elliptic restricted, and planetary three-body problems.

Orbital stability analysis of hypothetical Earth-mass and Luna-mass moons in the Sagarmatha (HD 100777) star system

Abstract Six of the solar system planets have 150 confirmed moons (Earth—1, Mars—2, Jupiter—53, Saturn—53, Uranus—27, Neptune—14) and seven of them (Ganymede, Titan, Callisto, Io, the Moon, Europa, Triton) have masses >0.001 $M_\oplus$. However, no exomoons have yet been discovered despite the successful detection of ∼5000 exoplanets. We can infer, based on the solar system planets, that these exoplanets are capable of hosting one or more exomoons. In this paper, we study the possible existence of hypothetical Earth-mass and Luna-mass moons orbiting the Jupiter-mass planet, Laligurans (HD 100777b) in the Sagarmatha (HD 100777) star system by means of orbital stability. We apply long-term orbital integrations and the MEGNO (mean exponential growth of nearby orbits) chaos indicator to study the orbital stability of the moons and predict a phase-space region comprising periodic, chaotic, and unstable orbits. The phase spaces primarily constitute the moon's semimajor axis, which extends from the host planet's Roche radius to the Hill radius, and full range of eccentricity. Specific points are picked from three different regions of the MEGNO map and run as single-orbit integration for up to 10 billion periods of the innermost orbit. Furthermore, the lifetime and maximum eccentricity maps are generated from the direct integration to inspect the stable and unstable orbital configurations. The analyses of these maps, with the aid of time-series plots, show that both moons maintain stable orbits in the low-eccentricity regime and semimajor axis between the Roche limit and 28.4% of the Hill radius of the planet.

Maneuver Strategy for Active Spacecraft to Avoid Space Debris and Return to the Original Orbit

During normal operation of the on-orbit spacecraft, if some satellite in a nearby orbit suddenly breaks apart, its debris will threat the safe operation of the on-orbit spacecraft. Therefore, it is necessary to study the active spacecraft’s avoidance of the space debris group and returning to the original orbit. In this way, the safe operation of on-orbit spacecraft will be guaranteed. However, as the geometric structure of the space debris group is constantly changing, it is hard to accurately demonstrate the changing shape of the debris group, let alone determine the unreachable domain. Traditional obstacle avoidance problems involve low speed of the vehicle; so, the application of artificial potential field and particle swarm algorithms is suitable for such problems. However, these two methods are not applicable to the maneuver strategy of spacecraft with high initial velocity. Therefore, to help spacecraft avoid the space debris group, a new method is required. This paper has established a simplified model to simulate the unreachable domain of the space debris group. It has modified the artificial potential field (APF) method and particle swarm optimization algorithm, with an aim to help spacecraft avoid the space debris group and return to the original orbit. Based on the method, the paper has proposed a three-stage maneuver strategy for the spacecraft to avoid the debris. To show the effectiveness of the method, this paper has simulated an on-orbit spacecraft’s avoidance of the space debris group nearby and returning to its original orbit. Through simulation, the feasibility of the maneuver strategy for spacecraft in the geosynchronous orbit is evaluated. The simulation results show that the method proposed in this paper can effectively accomplish the task.

Novel Pehlivan–Uyarŏglu Chaotic System Variants and their CFOA Based Realization

In this paper, four variants of Pehlivan–Uyarŏglu chaotic system (PUCS) have been proposed. The properties of the proposed PUCSs are examined through numerical simulations and the parameter values are obtained by observing the bifurcation diagrams for state variables. Further, the convergence/divergence of nearby orbits is investigated by noticing the evolution of Lyapunov exponents with time. It is found that the values of Lyapunov exponents are negative, zero and positive for all proposed variants thus confirming the chaoticity of the proposals. The strangeness of the proposed variants is also studied. The stability of PUCS and its proposed variants is examined using Jacobi stability analysis. A current feedback operational amplifier (CFOA) based circuit is put forward that can realize the existing PUCS and its proposed variants, by simply adjusting the component values. The proposed realization is compact (23% saving in overall component count) in comparison to its operational amplifier (OpAmp) based counterpart. The behavior of the proposed variants in time domain, frequency domain and phase space have been examined through simulations in LTspice design environment. Furthermore, the feasibility of the proposed variants is also discussed through presenting the electronic circuit implementation of two of the variants and the results obtained are in good agreement with the LTspice simulations. Monte Carlo (MC) simulations are also included to show the robustness of the proposed circuit against parameter variations.

Pearls on a String: Numerous Stellar Clusters Strung Along the Same Orbit

Abstract Stars originate from the dense interstellar medium, which exhibits filamentary structure to scales of ∼1 kpc in galaxies like our Milky Way. We explore quantitatively how much resulting large-scale correlation there is among different stellar clusters and associations in orbit-phase space, characterized here by actions and angles. As a starting point, we identified 55 prominent stellar overdensities in the 6D space of orbit (actions) and orbit-phase (angles) among the ∼2.8 million stars with radial velocities from Gaia EDR3 and with d ≤ 800 pc. We then explored the orbit-phase distribution of all sample stars in the same orbit patch as any one of these 55 overdensities. We find that very commonly numerous other distinct orbit-phase overdensities exist along these same orbits, like pearls on a string. These “pearls” range from known stellar clusters to loose, unrecognized associations. Among orbit patches defined by one initial orbit-phase overdensity, 50% contain at least 8 additional orbit-phase pearls of 10 cataloged members; 20% of them contain 20 additional pearls. This is in contrast to matching orbit patches sampled from a smooth mock catalog, or offset nearby orbit patches, where there are only 2 (or 5, respectively) comparable pearls. Our findings quantify for the first time how common it is for star clusters and associations to form at distinct orbit-phases of nearly the same orbit. This may eventually offer a new way to probe the 6D orbit structure of the filamentary interstellar medium.

Signature splitting of the g7/2[404]7/2+ bands in Ba131 and Ce133

Excited states in Ba131 and Ce133 were studied using in-beam γ-ray spectroscopy through the Sn122(C13,4n)Ba131 and Te125(C12,4n)Ce133 reactions, respectively. A strongly coupled band, associated with the νg7/2[404]7/2+ configuration, was identified in Ba131 and Ce133. It is the first time to observe the νg7/2[404]7/2+ bands in the N=75 isotones. The signature partners exhibit considerable energy splitting in comparison with those in the πg7/2[404]7/2+ bands in the odd-A Ta and Re isotopes. Extensive cranked shell model and quasiparticle-plus-triaxial-rotor model calculations reveal the origin of the signature splitting, which depends not only on the triaxiality, but also on the configuration mixing with nearby low-j orbitals.3 MoreReceived 30 July 2021Revised 13 October 2021Accepted 19 November 2021DOI:https://doi.org/10.1103/PhysRevC.104.064304©2021 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCollective levelsElectromagnetic transitionsNuclear structure & decaysTransfer reactionsProperties90 ≤ A ≤ 149Nuclear Physics

Revisiting TrES-5 b: departure from a linear ephemeris instead of short-period transit timing variation

Aims. The orbital motion of the transiting hot Jupiter TrES-5 b was reported to be perturbed by a planetary companion on a nearby orbit. Such compact systems do not frequently occur in nature, and investigating their orbital architecture could shed some light on the formation processes of hot Jupiters. Methods. We acquired 15 new precise photometric time-series for 12 transits of TrES-5 b between June 2019 and October 2020 using 0.9–2.0 m telescopes. The method of precise transit timing was employed to verify the deviation of the planet from the Keplerian motion. Results. Although our results show no detectable short-time variation in the orbital period of TrES-5 b and the existence of the additional nearby planet is not confirmed, the new transits were observed about two minutes earlier than expected. We conclude that the orbital period of the planet could vary on a long timescale. We found that the most likely explanation of the observations is the line-of-sight acceleration of the system’s barycentre caused by the orbital motion induced by a massive, wide-orbiting companion.

Pluto’s Resonant Orbit Visualized in 4D

Abstract Orbital resonance phenomena are notoriously difficult to communicate in words due to the complex dynamics arising from the interplay of gravity and orbital angular momentum. A well known example is Pluto’s 3:2 mean motion resonance with Neptune. We have developed a python software tool to visualize the full three-dimensional aspects of Pluto’s resonant orbital dynamics over time. The visualizations include still images and animated movies. By contrasting Pluto’s resonant dynamics with the dynamics of a nearby non-resonant orbit, this tool enables better understanding and exploration of complex planetary dynamics phenomena.

Asymptotic expansivity

We study continuous maps of metric spaces for which two nearby orbits are asymptotic (termed asymptotically expansive for short). We also analyze the bi-asymptotically expansive homeomorphisms namely asymptotically expansive homeomorphisms with asymptotically expansive inverse. Indeed, we obtain necessary and sufficient conditions for asymptotic expansivity, characterize the asymptotically expansive homeomorphisms of the circle, prove a spectral decomposition theorem and estimate the entropy through the growth rate of the periodic orbits.

Energetics of collapsible channel flow with a nonlinear fluid-beam model

We consider flow along a finite-length collapsible channel driven by a fixed upstream flux, where a section of one wall of a planar rigid channel is replaced by a plane-strain elastic beam subject to uniform external pressure. A modified constitutive law is used to ensure that the elastic beam is energetically conservative. We apply the finite element method to solve the fully nonlinear steady and unsteady systems. In line with previous studies, we show that the system always has at least one static solution and that there is a narrow region of the parameter space where the system simultaneously exhibits two stable static configurations: an (inflated) upper branch and a (collapsed) lower branch, connected by a pair of limit point bifurcations to an unstable intermediate branch. Both upper and lower static configurations can each become unstable to self-excited oscillations, initiating either side of the region with multiple static states. As the Reynolds number increases along the upper branch the oscillatory limit cycle persists into the region with multiple steady states, where interaction with the intermediate static branch suggests a nearby homoclinic orbit. These oscillations approach zero amplitude at the upper branch limit point, resulting in a stable tongue between the upper and lower branch oscillations. Furthermore, this new formulation allows us to calculate a detailed energy budget over a period of oscillation, where we show that both upper and lower branch instabilities require an increase in the work done by the upstream pressure to overcome the increased dissipation.