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We consider a system in which both the parent star and the Earth-like exoplanet move on circular orbits. Using numerical methods, such as the orbit classification technique, we study all types of trajectories of possible exomoons around the exoplanet. In particular, we scan the phase space around the exoplanet and we distinguish between bounded, collisional, and escaping trajectories, considering both retrograde and prograde types of motion. In the case of bounded regular motion, we also use the grid method and a standard predictor-corrector procedure for revealing the corresponding network of symmetric periodic solutions, while we also compute their linear stability.

We numerically investigate the motion of stars on the meridional plane of an axially symmetric disk galaxy model, containing a central supermassive black hole, represented by the Paczynski-Wiita potential. By using this pseudo-Newtonian potential we can replicate important relativistic properties, such as the existence of the Schwarzschild radius. After classifying extensive samples of initial conditions of trajectories, we manage to distinguish between collisional, ordered, and chaotic motion. Besides, all starting conditions of regular orbits are further classified into families of regular orbits. Our results are presented through modern color-coded basin diagrams on several types of two-dimensional planes. Our analysis reveals that both the mass of the black hole (in direct relation with the Schwarzschild radius) as well as the angular momentum play an important role in the character of orbits of stars. More specifically, the trajectories of low angular momentum stars are highly affected by the mass of the black hole, while high angular momentum stars seem to be unaffected by the central black hole. Comparison with previous related outcomes, using Newtonian potentials for the central region of the galaxy, is also made., 10 pages, 10 figures

In this paper, we consider a Newtonian system whose relativistic counterpart describes a superimposed halo with a black hole. Our aim is to determine how the quadrupole and octupole moments affect the nature of the motion of a test particle, moving in the close vicinity of the black hole. The different types of trajectories for the test particle are mainly classified as bounded, collisional, and escaping, by using modern color-coded basin diagrams. Moreover, an additional analysis is carried out for distinguishing between the different types of bounded motion (regular, sticky, and chaotic). Our results strongly indicate that the multipole moments, along with the total orbital energy, highly affect the final state of the test particle, while at the same time the basin geometry of the phase space tends to be highly dominated by collision and escape orbits., 10 pages, 7 figures

Continuing the series of papers on a new model for a barred galaxy, we investigate the heteroclinic connections between the two normally hyperbolic invariant manifolds sitting over the two index-1 saddle points of the effective potential. The heteroclinic trajectories and the nearby periodic orbits of similar shape populate the bar region of the galaxy and a neighbourhood of its nucleus. Thereby we see a direct relation between the important structures of the interior region of the galaxy and the projection of the heteroclinic tangle into the position space. As a side result, we obtain a detailed picture of the primary heteroclinic intersection surface in the phase space., Comment: Published in Monthly Notices of the Royal Astronomical Society (MNRAS) journal

This article aims to investigate the points of equilibrium and the associated convergence basins in a seventh-order generalized H\'enon-Heiles potential. Using the well-known Newton-Raphson iterator we numerically locate the position of the points of equilibrium, while we also obtain their linear stability. Furthermore, we demonstrate how the two variable parameters, entering the generalized H\'enon-Heiles potential, affect the convergence dynamics of the system as well as the fractal degree of the basin diagrams. The fractal degree is derived by computing the (boundary) basin entropy as well as the uncertainty dimension., Comment: 11 pages, 8 figures

We explore a pseudo-Newtonian planar circular restricted three-body problem in which the primaries are modeled using an approximate gravitational potential up to the second nonvanishing term of the Fodor–Hoenselaers–Perjés expansion. We aim to understand how the main free parameters of the system affect its dynamical properties. In particular, we determine how the mass of the primaries as well as the transition parameters affect not only the properties of the points of equilibrium (total number, locations, and linear stability) but also the networks of simple symmetric periodic orbits. Our results show that, under this approach, significant variations are observed in the fixed points (number and stability) and periodic orbits of the planar circular restricted three-body problem, even when small contributions of the non-Newtonian terms are considered. We also provide direct applications of the new model potential in real observable binary stellar systems.

The size distribution and orbital architecture of dust, grains, boulders, asteroids, and major planets during the giant branch phases of evolution dictate the preponderance and observability of the eventual debris, which have been found to surround white dwarfs and pollute their atmospheres with metals. Here, we utilize the photogravitational planar restricted three-body problem in one-planet giant branch systems in order to characterize the orbits of grains as the parent star luminosity and mass undergo drastic changes. We perform a detailed dynamical analysis of the character of grain orbits (collisional, escape, or bounded) as a function of location and energy throughout giant branch evolution. We find that for stars with main-sequence masses of $2.0M_{\odot}$, giant branch evolution, combined with the presence of a planet, ubiquitously triggers escape in grains smaller than about 1 mm, while leaving grains larger than about 5 cm bound to the star. This result is applicable for systems with either a terrestrial or giant planet, is largely independent of the location of the planet, and helps establish a radiative size threshold for escape of small particles in giant branch planetary systems., Comment: Published in Astronomy & Astrophysics journal (A&A)

Nearly all known white dwarf planetary systems contain detectable rocky debris in the stellar photosphere. A glaring exception is the young and still evolving white dwarf WD J0914+1914, which instead harbours a giant planet and a disc of pure gas. The stability boundaries of this disc and the future prospects for this white dwarf to be polluted with rocks depend upon the mass and orbit of the planet, which are only weakly constrained. Here we combine an ensemble of plausible planet orbits and masses to determine where observers should currently expect to find the outer boundary of the gas disc. We do so by performing a sweep of the entire plausible phase space with short-term numerical integrations. We also demonstrate that particle-star collisional trajectories, which would lead to the (unseen) signature of rocky metal pollution, occupy only a small fraction of the phase space, mostly limited to particle eccentricities above 0.75. Our analysis reveals that a highly inflated planet on a near-circular orbit is the type of planet which is most consistent with the current observations., Comment: Accepted for publication in Monthly Notices of the Royal Astronomical Society Main Journal

The scope of this work is to perform a numerical investigation of the orbital dynamics for a test particle in the pseudo-Newtonian Hill problem. Large two-dimensional sets of initial conditions of prograde and retrograde orbits are investigated. The orbits are classified as bounded (chaotic, sticky or regular), escaping and collision orbits. The smaller alignment index (SALI) method is used to identify chaotic orbits. Additionally, the influence of the energy (or equivalently the value of the Jacobi constant) and of the Schwarzschild radius on the orbital structure of the system are determined. Our numerical results are compared with related previous ones, corresponding to the classical version of the Hill problem.

The dynamics of the pseudo-Newtonian restricted four-body problem has been studied in the present paper, where the primaries have equal masses. The parametric variation of the existence as well as the position of the libration points are determined, when the value of the transition parameter $\epsilon \in [0, 1]$. The stability of these libration points has also been discussed. Our study reveals that the Jacobi constant as well as transition parameter $\epsilon$ have substantial effect on the regions of possible motion, where the fourth body is free to move. The multivariate version of Newton-Raphson iterative scheme is introduced for determining the basins of attraction in the configuration $(x,y)$ plane. A systematic numerical investigation is executed to reveal the influence of the transition parameter on the topology of the basins of convergence. In parallel, the required number of iterations is also noted to show its correlations to the corresponding basins of convergence. It is unveiled that the evolution of the attracting regions in the pseudo-Newtonian restricted four-body problem is a highly complicated yet worth studying problem., Comment: Published in New Astronomy journal

The escape mechanism of orbits in a star cluster rotating around its parent galaxy in a circular orbit is investigated. A three degrees of freedom model is used for describing the dynamical properties of the Hamiltonian system. The gravitational field of the star cluster is represented by a smooth and spherically symmetric Plummer potential. We distinguish between ordered and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. The Smaller Alignment Index (SALI) method is used for determining the regular or chaotic nature of the orbits. The basins of escape are located and they are also correlated with the corresponding escape time of the orbits. Areas of bounded regular or chaotic motion and basins of escape were found to coexist in the $(x,z)$ plane. The properties of the normally hyperbolic invariant manifolds (NHIMs), located in the vicinity of the index-1 Lagrange points $L_1$ and $L_2$, are also explored. These manifolds are of paramount importance as they control the flow of stars over the saddle points, while they also trigger the formation of tidal tails observed in star clusters. Bifurcation diagrams of the Lyapunov periodic orbits as well as restrictions of the Poincar\'e map to the NHIMs are deployed for elucidating the dynamics in the neighbourhood of the saddle points. The extended tidal tails, or tidal arms, formed by stars with low velocity which escape through the Lagrange points are monitored. The numerical results of this work are also compared with previous related work., Comment: Published in Monthly Notices of the Royal Astronomical Society (MNRAS) journal

A three degrees of freedom (3-dof) barred galaxy model composed of a spherically symmetric nucleus, a bar, a flat disc and a spherically symmetric dark matter halo is used for investigating the dynamics of the system. We use colour-coded plots to demonstrate how the value of the semi-major axis of the bar influences the regular or chaotic dynamics of the 3-dof system. For distinguishing between ordered and chaotic motion we use the Smaller ALingment Index (SALI) method, a fast yet very accurate tool. Undoubtedly, the most important elements of the dynamics are the normally hyperbolic invariant manifolds (NHIMs) located in the vicinity of the index 1 Lagrange points $L_2$ and $L_3$. These manifolds direct the flow of stars over the saddle points, while they also trigger the formation of rings and spirals. The dynamics in the neighbourhood of the saddle points is visualized by bifurcation diagrams of the Lyapunov orbits as well as by the restriction of the Poincar\'e map to the NHIMs. In addition, we reveal how the semi-major axis of the bar influences the structure of these manifolds which determine the final stellar structure (rings or spirals). Our numerical simulations suggest that in galaxies with weak bars the formation of $R_1$ rings or $R_1'$ pseudo-rings is favoured. In the case of galaxies with intermediate and strong bars the invariant manifolds seem to give rise to $R_1R_2$ rings and twin spiral formations, respectively. We also compare our numerical outcomes with earlier related work and with observational data., Comment: Published in Monthly Notices of the Royal Astronomical Society (MNRAS) journal

The three-dimensional version of the circular restricted problem of three bodies is utilized to describe a system comprising a host star and an exoplanet. The third body, playing the role of a test particle, can be a comet or an asteroid, or even a small exomoon. Combining the grid classification method with two-dimensional color-coded basin maps, we determine the nature of the motion of the test particle by distinguishing between collision, escaping, and bounded motion. In the case of ordered bounded motion, we also obtain the orientation (retrograde or prograde) as well as the geometry (circulating around one or both of the two main bodies) of the trajectories of the third body, which starts from either the pericenter or apocenter. Following this approach, we are able to systematically explore the dependence of the motion type of the test particle on the initial values of the semimajor axis, eccentricity, and inclination of its orbit.

The on-plane version of the restricted problem of 3 bodies with a continuation fraction potential is numerically investigated. The idea is to consider that one of the primaries is a radiation source and the secondary one is not spherical. By adopting the grid classification method we locate the coordinates, on the X Y − plane, of all the points of equilibrium, for several values of the involved parameters. The stability of the libration points is also computed, as a function of the same parameters. The shape as well as the properties of the Newton–Raphson basins of convergence, associated with the equilibria of the system, are also explored for obtaining the optimal starting conditions of the iterator. Our analysis reveals that the new potential has additional points of equilibrium, with respect to the classical 3-body problem.

The dynamics of the orbital motion in the planar elliptic restricted three-body problem are investigated, by using the method of grid classification. In this system, the secondary body is an exoplanet, while the corresponding primary body is its parent star. We numerically investigate how several dynamical quantities of the system, such as the orbital energy, the eccentricity, the true anomaly, and the mass parameter, influence several aspects of the motion of the test particle, such as the final state as well as the time of escape/collision of the orbits. Color-coded basin diagrams are utilized for displaying all the different types of basins, using two-dimensional maps. The results of this analysis are then compared to similar ones from the classical version of the circular problem of three bodies.

We examine the orbital dynamics of the planar pseudo-Newtonian Copenhagen problem, in the case of a binary system of Schwarzschild-like primaries, such as super-massive black holes. In particular, we investigate how the Jacobi constant (which is directly connected with the energy of the orbits) influences several aspects of the orbital dynamics, such as the final state of the orbits. We also determine how the relativistic effects (i.e., the Schwarzschild radius) affect the character of the orbits, by comparing our results with the classical Newtonian problem. Basin diagrams are deployed for presenting all the different basin types, using multiple types of planes with two dimensions. We demonstrate that both the Jacobi constant as well as the Schwarzschild radius highly influence the character of the orbits, as well as the degree of fractality of the dynamical system., Comment: Published in Monthly Notices of the Royal Astronomical Society (MNRAS) journal

The circular version of the restricted three-body problem, along with the method of grid classification are used to determine the character of the trajectories of a test particle, which move in a binary exoplanetary system. The binary system can be either a parent star-exoplanet or an exoplanet–exoplanet or exomoon, while the test particle is considered to be an asteroid or comet, a space probe, or even a small exomoon in the case where the primary body is a star. By using modern two-dimensional color maps, we succeeded in classifying the starting conditions and distinguishing between bounded, escaping, and collision type of motion for the test particle. Furthermore, in the case of bounded regular motion, we further classify the starting conditions by considering their geometry (revolving around one or both main bodies) and orientation (prograde or retrograde, with respect to a rotating coordinate system of the primaries). For the initial setup of the test particle we consider two starting conditions: the launch from pericenter or apocenter. The final states are qualitatively visualized through two-dimensional basin diagrams. This approach allowed us to systematically investigate and extract dynamical information on the dependency of the test particle final state as a function of the particle’s initial semi-major axis and eccentricity for a given primary and secondary mass ratio. Finally, we applied the restricted three-body model on several exoplanetary systems with observed mass-ratios and studied the dynamical behavior of a test-mass.

The dynamics of a test particle in a non-spinning binary black hole system of equal masses is numerically investigated. The binary system is modeled in the context of the pseudo-Newtonian circular restricted three-body problem, such that the primaries are separated by a fixed distance and move in a circular orbit around each other. In particular, the Paczy\'{n}ski-Wiita potential is used for describing the gravitational field of the two non-Newtonian primaries. The orbital properties of the test particle are determined through the classification of the initial conditions of the orbits, using several values of the Jacobi constant, in the Hill's regions of possible motion. The initial conditions are classified into three main categories: (i) bounded, (ii) escaping and (iii) displaying close encounters. Using the smaller alignment index (SALI) chaos indicator, we further classify bounded orbits into regular, sticky or chaotic. To gain a complete view of the dynamics of the system, we define grids of initial conditions on different types of two-dimensional planes. The orbital structure of the configuration plane, along with the corresponding distributions of the escape and collision/close encounter times, allow us to observe the transition from the classical Newtonian to the pseudo-Newtonian regime. Our numerical results reveal a strong dependence of the properties of the considered basins with the Jacobi constant as well as with the Schwarzschild radius of the black holes., Comment: Published in Monthly Notices of the Royal Astronomical Society (MNRAS) journal

The circular Sitnikov problem, where the two primary bodies are prolate or oblate spheroids, is numerically investigated. In particular, the basins of convergence on the complex plane are revealed by using a large collection of numerical methods of several order. We consider four cases, regarding the value of the oblateness coefficient which determines the nature of the roots (attractors) of the system. For all cases we use the iterative schemes for performing a thorough and systematic classification of the nodes on the complex plane. The distribution of the iterations as well as the probability and their correlations with the corresponding basins of convergence are also discussed. Our numerical computations indicate that most of the iterative schemes provide relatively similar convergence structures on the complex plane. However, there are some numerical methods for which the corresponding basins of attraction are extremely complicated with highly fractal basin boundaries. Moreover, it is proved that the efficiency strongly varies between the numerical methods., Comment: Published in Astrophysics and Space Science (A&SS) journal

The aim of this work is to compare the orbital dynamics in three different models describing the properties of a star cluster rotating around its parent galaxy in a circular orbit. In particular, we use the isochrone and the Hernquist potentials to model the spherically symmetric star cluster and we compare our results with the corresponding ones of a previous work in which the Plummer model was applied for the same purpose. Our analysis takes place both in the configuration $(x,y)$ and in the phase $(x,\dot{x})$ space in order to elucidate the escape process as well as the overall orbital properties of the tidally limited star cluster. We restrict our investigation into two dimensions and we conduct a thorough numerical analysis distinguishing between ordered and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels above the critical escape energy. It is of particular interest to determine the escape basins towards the two exit channels (near the Lagrangian points $L_1$ and $L_2$) and relate them with the corresponding escape times of the orbits., Comment: Published in MNRAS journal. arXiv admin note: previous papers with related context: arXiv:1411.4864, arXiv:1505.03968, arXiv:1505.03847

We numerically explore the Newton-Raphson basins of convergence, related to the libration points (which act as attractors), in the planar circular restricted five-body problem (CR5BP). The evolution of the position and the linear stability of the equilibrium points is determined, as a function of the value of the mass parameter. The attracting regions, on several types of two dimensional planes, are revealed by using the multivariate version of the classical Newton-Raphson iterative method. We perform a systematic investigation in an attempt to understand how the mass parameter affects the geometry as well as the degree of fractality of the basins of attraction. The regions of convergence are also related with the required number of iterations and also with the corresponding probability distributions., Published in Astrophysics and Space Science (A&SS) journal. Previous paper with related context: arXiv:1801.00710

The Newton-Raphson basins of attraction, associated with the libration points (attractors), are revealed in the pseudo-Newtonian planar circular restricted three-body problem, where the primaries have equal masses. The parametric variation of the position as well as of the stability of the equilibrium points is determined, when the value of the transition parameter $\epsilon$ varies in the interval $[0,1]$. The multivariate Newton-Raphson iterative scheme is used to determine the attracting domains on several types of two-dimensional planes. A systematic and thorough numerical investigation is performed in order to demonstrate the influence of the transition parameter on the geometry of the basins of convergence. The correlations between the basins of attraction and the corresponding required number of iterations are also illustrated and discussed. Our numerical analysis strongly indicates that the evolution of the attracting regions in this dynamical system is an extremely complicated yet very interesting issue., Comment: Published in Astrophysics and Space Science (A&SS) journal. Previous papers with related context: arXiv:1702.07279, arXiv:1706.07044, arXiv:1704.02273, arXiv:1709.06631

The basins of convergence, associated with the roots (attractors) of a complex equation, are revealed in the Hill problem with oblateness and radiation, using a large variety of numerical methods. Three cases are investigated, regarding the values of the oblateness and radiation. In all cases, a systematic and thorough scan of the complex plane is performed in order to determine the basins of attraction of the several iterative schemes. The correlations between the attracting domains and the corresponding required number of iterations are also illustrated and discussed. Our numerical analysis strongly suggests that the basins of convergence, with the highly fractal basin boundaries, produce extraordinary and beautiful formations on the complex plane., Published in Astrophysics and Space Science (A&SS) journal

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.

The aim of this work is to explore the escape process of three-dimensional orbits in a star cluster rotating around its parent galaxy in a circular orbit. The gravitational field of the cluster is represented by a smooth, spherically symmetric Plummer potential, while the tidal approximation was used to model the steady tidal field of the galaxy. We conduct a thorough numerical analysis distinguishing between regular and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. It is of particular interest to locate the escape basins towards the two exit channels and relate them with the corresponding escape times of the orbits. For this purpose, we split our investigation into three cases depending on the initial value of the $z$ coordinate which was used for launching the stars. The most noticeable finding is that the majority of stars initiated very close to the primary $(x,y)$ plane move in chaotic orbits and they remain trapped for vast time intervals, while orbits with relatively high values of $z_0$ on the other hand, form well-defined basins of escape. It was also observed, that for energy levels close to the critical escape energy the escape rates of orbits are large, while for much higher values of energy most of the orbits have low escape periods or they escape immediately to infinity. We hope our outcomes to be useful for a further understanding of the dissolution process and the escape mechanism in open star clusters., Comment: Published in MNRAS journal. arXiv admin note: previous paper with related context: arXiv:1404.4285

A three-dimensional dynamical model for a galaxy hosting a BL Lacertae object is constructed. The model consists of a logarithmic potential representing an elliptical host galaxy with a bulge of radius cb and a dense massive nucleus. Using numerical experiments, we try to distinguish between regular and chaotic motion in both 2D and 3D system. In particular, we investigate how the basic parameters of our model, such as the mass of the nucleus, the internal perturbation and the flattening parameters influence the amount and the degree of chaos. Interesting correlations are presented for both 2D and 3D dynamical models. Our numerical results are explained and supported using elementary theoretical arguments and analytical calculations. Of particular interest is the local integral of motion which have been found to exist in the vicinity of stable periodic points. The obtained numerical outcomes of the present research are linked and also compared with several data derived from observations. (© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The main objective of this work is to determine the character of orbits of stars moving in the meridional $(R,z)$ plane of an axially symmetric time-independent disk galaxy model with a central massive nucleus and an additional spherical dark matter halo component. In particular, we try to reveal the influence of the scale length of the dark matter halo on the different families of orbits of stars, by monitoring how the percentage of chaotic orbits, as well as the percentages of orbits of the main regular resonant families evolve when this parameter 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 safely 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. Our numerical computations reveal that when the dark matter halo is highly concentrated, that is when the scale length has low values the vast majority of star orbits move in regular orbits, while on the other hand in less concentrated dark matter halos the percentage of chaos increases significantly. We also compared our results with early related work., Comment: Published in Baltic Astronomy journal. arXiv admin note: previous papers with related context: arXiv:1404.4194, arXiv:1404.3961, arXiv:1309.5607

The aim of this paper is to numerically investigate the orbital dynamics of a test particle, in the planar circular restricted Pluto-Charon system. By numerically integrating the equations of motion, forward in time, with several large sets of initial conditions of orbits, we manage to classify them into three main categories: (i) bounded (regular or chaotic) (ii) escaping and (iii) collision orbits. The SALI method is used for safely identifying the chaotic or regular nature of the orbits. Furthermore, we determine the influence of the value of the total energy (or equivalently the value of the Jacobi constant) on the orbital structure of the system. In addition, the network as well as the stability of the symmetric periodic orbits are also revealed. In our analysis, we consider a large variety of symmetric periodic orbits, regarding their multiplicity.

We use a simple dynamical model in order to investigate the regular or chaotic character of orbits in a barred galaxy with a central, spherically symmetric, dense nucleus and a flat disk. In particular, we explore how the total orbital energy influences the overall orbital structure of the system, by computing in each case the percentage of regular, sticky and chaotic orbits. In an attempt to distinguish safely and with certainty between ordered and chaotic motion, we apply the Smaller ALingment Index (SALI) as a chaos detector to extensive samples of orbits obtained by integrating numerically the basic equations of motion as well as the variational equations. We integrate large sets of initial conditions of orbits in several types of two dimensional planes for better understanding of the orbital properties. Our numerical calculations suggest, that the value of the energy has a huge impact on the percentages of the orbits, thus indicating that a rotating barred galaxy is indeed a very interesting stellar quantity., Comment: Published in Astronomische Nachrichten (AN) journal. arXiv admin note: previous papers with related context: arXiv:1506.01496, arXiv:1505.03968, arXiv:1502.02510, arXiv:1604.04622, arXiv:1604.04613

The planar equilateral restricted four-body problem where two of the primaries have equal masses is used in order to determine the Newton-Raphson basins of convergence associated with the equilibrium points. The parametric variation of the position of the libration points is monitored when the value of the mass parameter $m_3$ varies in predefined intervals. The regions on the configuration $(x,y)$ plane occupied by the basins of attraction are revealed using the multivariate version of the Newton-Raphson iterative scheme. The correlations between the attracting domains of the equilibrium points and the corresponding number of iterations needed for obtaining the desired accuracy are also illustrated. We perform a thorough and systematic numerical investigation by demonstrating how the dynamical parameter $m_3$ influences the shape, the geometry and the degree of fractality of the converging regions. Our numerical outcomes strongly indicate that the mass parameter is indeed one of the most influential factors in this dynamical system., Published in Astrophysics and Space Science (A&SS) journal

In this paper we use the planar circular restricted three-body problem where one of the primary bodies is an oblate spheroid or an emitter of radiation in order to determine the basins of attraction associated with the equilibrium points. The evolution of the position of the five Lagrange points is monitored when the values of the mass ratio $\mu$, the oblateness coefficient $A_1$, and the radiation pressure factor $q$ vary in predefined intervals. The regions on the configuration $(x,y)$ plane occupied by the basins of attraction are revealed using the multivariate version of the Newton-Raphson method. The correlations between the basins of convergence of the equilibrium points and the corresponding number of iterations needed in order to obtain the desired accuracy are also illustrated. We conduct a thorough and systematic numerical investigation demonstrating how the dynamical quantities $\mu$, $A_1$, and $q$ influence the basins of attractions. Our results suggest that the mass ratio and the radiation pressure factor are the most influential parameters, while on the other hand the structure of the basins of convergence are much less affected by the oblateness coefficient., Comment: Published in Astrophysics and Space Science (A&SS) journal

We use a simple dynamical model which consists of a harmonic oscillator and a spherical component, in order to investigate the regular or chaotic character of orbits in a barred galaxy with a central spherically symmetric nucleus. Our aim is to explore how the basic parameters of the galactic system influence the nature of orbits, by computing in each case the percentage of chaotic orbits, as well as the percentages of different types of regular orbits. We also give emphasis to the types of regular orbits that support either the formation of nuclear rings or the barred structure of the galaxy. We provide evidence that the traditional x1 orbital family does not always dominate in barred galaxy models since we found several other types of resonant orbits which can also support the barred structure. We also found that sparse enough nuclei, fast rotating bars and high energy models can support the galactic bars. On the other hand, weak bars, dense central nuclei, slowly rotating bars and low energy models favor the formation of nuclear rings. We also compare our results with previous related work., Published in Research in Astronomy and Astrophysics (RAA) journal

The orbital dynamics of a spacecraft, or a comet, or an asteroid in the Earth-Moon system in a scattering region around the Moon using the three dimensional version of the circular restricted three-body problem is numerically investigated. The test particle can move in bounded orbits around the Moon or escape through the openings around the Lagrange points $L_1$ and $L_2$ or even collide with the surface of the Moon. We explore in detail the first four of the five possible Hill's regions configurations depending on the value of the Jacobi constant which is of course related with the total orbital energy. We conduct a thorough numerical analysis on the phase space mixing by classifying initial conditions of orbits in several two-dimensional types of planes and distinguishing between four types of motion: (i) ordered bounded, (ii) trapped chaotic, (iii) escaping and (iv) collisional. In particular, we locate the different basins and we relate them with the corresponding spatial distributions of the escape and collision times. Our outcomes reveal the high complexity of this planetary system. Furthermore, the numerical analysis suggests a strong dependence of the properties of the considered basins with both the total orbital energy and the initial value of the $z$ coordinate, with a remarkable presence of fractal basin boundaries along all the regimes. Our results are compared with earlier ones regarding the planar version of the Earth-Moon system., Published in Astrophysics and Space Science (A&SS) journal. arXiv admin note: previous papers with related context: arXiv:1512.08683, arXiv:1508.05201

We study the behavior of orbits in two different galactic dynamical models, describing the motion in the central parts of a triaxial elliptical galaxy with a dense nucleus. Numerical experiments show that both models display regular motion together with extended chaotic regions. A detailed investigation of the properties of motion is made for the 2D and 3D Hamiltonian systems, using a number of different dynamical parameters, such as the Poincare surface of section, the maximal Lyapunov Characteristic Exponent, the S(c) spectrum, the S(w) spectrum and the P(f) indicator. The numerical calculations suggest that the properties of motion in both potentials are very similar. Our results show that one may use different kinds of gravitational potentials in order to describe the motion in triaxial galaxies while obtaining quantitatively similar results., Published in Research in Astronomy and Astrophysics journal

In the present article, we present a new gravitational galactic model, describing motion in elliptical as well as in disk galaxies, by suitably choosing the dynamical parameters. Moreover, a new dynamical parameter, the S(g) spectrum, is introduced and used, in order to detect islandic motion of resonant orbits and the evolution of the sticky regions. We investigate the regular or chaotic character of motion, with emphasis in the different dynamical models and make an extensive study of the sticky regions of the system. We use the classical method of the Poincare (r-pr) phase plane and the new dynamical parameter of the S(g) spectrum. The LCE is used, in order to make an estimation of the degree of chaos in our galactic model. In both cases, the numerical calculations, suggest that our new model, displays a wide variety of families of regular orbits, compared to other galactic models. In addition to the regular motion, this new model displays also chaotic regions. Furthermore, the extent of the chaotic regions increases, as the value of the flatness parameter b of the model increases. Moreover, our simulations indicate, that the degree of chaos in elliptical galaxies, is much smaller than that in dense disk galaxies. In both cases numerical calculations show, that the degree of chaos increases linearly, as the flatness parameter b increases. In addition, a linear relationship between the critical value of angular momentum and the b parameter if found, in both cases (elliptical and disk galaxies). Some theoretical arguments to support the numerical outcomes are presented. Comparison with earlier work is also made., Published in New Astronomy journal

We study the regular or chaotic character of orbits in a 3D dynamical model, describing a triaxial galaxy surrounded by a spherical dark halo component. Our numerical experiments suggest that the percentage of chaotic orbits decreases exponentially as the mass of the dark halo increases. A linear increase of the percentage of the chaotic orbits was observed as the scale length of the halo component increases. In order to distinguish between regular and chaotic motion, we chose to use the total angular momentum Ltot of the 3D orbits as a new indicator. Comparison with other, previously used, dynamical indicators, such as the Lyapunov Characteristic Exponent or the P(f) spectral method, shows that the Ltot indicator gives very fast and reliable results for characterizing the nature of orbits in galactic dynamical models., Comment: Published in Research in Astronomy and Astrophysics (RAA) journal

The theory of the post-Newtonian (PN) planar circular restricted three-body problem is used for numerically investigating the orbital dynamics of a test particle (e.g. a comet, asteroid, meteor or spacecraft) in the planar Sun–Jupiter system with a scattering region around Jupiter. For determining the orbital properties of the test particle, we classify large sets of initial conditions of orbits for several values of the Jacobi constant in all possible Hill region configurations. The initial conditions are classified into three main categories: (i) bounded, (ii) escaping and (iii) collisional. Using the smaller alignment index (SALI) chaos indicator, we further classify bounded orbits into regular, sticky or chaotic. In order to get a spherical view of the dynamics of the system, the grids of the initial conditions of the orbits are defined on different types of two-dimensional planes. We locate the different types of basins and we also relate them with the corresponding spatial distributions of the escape and collision time. Our thorough analysis exposes the high complexity of the orbital dynamics and exhibits an appreciable difference between the final states of the orbits in the classical and PN approaches. Furthermore, our numerical results reveal a strong dependence of the properties of the considered basins with the Jacobi constant, along with a remarkable presence of fractal basin boundaries. Our outcomes are compared with the earlier ones regarding other planetary systems.

The ordered or chaotic character of orbits of stars moving in the meridional $(R,z)$ plane of an analytic axisymmetric time-independent disk galaxy model with an additional spherically symmetric central nucleus is investigated. Our aim is to determine how the total energy influences the orbital structure of the galaxy. For this purpose we monitor how the percentage of chaotic orbits as well as the rates of orbits composing the main regular families evolve as a function of the value of the energy. In order to distinguish with certainty between chaotic and ordered motion we use the SALI method in extensive sets of initial conditions of orbits. Moreover, a spectral method is applied for identifying the various regular families and also for recognizing the secondary resonances that bifurcate from them. Our numerical computations suggest that for low energy levels the observed amount of chaos is high and the orbital content is rather poor, while for high energy levels, corresponding to global motion, regular motion dominates and many secondary higher resonances emerge. We also compared our results with previous related work., Comment: Published in Baltic Astronomy journal. arXiv admin note: previous paper with related context: arXiv:1501.06699

Locating the position of periodic orbits in galaxies is undoubtedly an issue of paramount importance. We reveal the position and the stability of periodic orbits of stars moving in the meridional plane $(R,z)$ of an axially symmetric galactic model with a disk, a spherical nucleus, and a biaxial dark matter halo component. In particular, we study how all the involved parameters of the dynamical system influence the position and the stability of all resonant families. To locate the position and measure the stability of periodic orbits we use a highly sensitive numerical code which is able to identify resonant periodic orbits of the type $n:m$. Two cases are studied for every parameter: (i) the case where the dark matter halo component is prolate and (ii) the case where an oblate dark matter halo is present. Our numerical exploration reveals that all the dynamical quantities affect, more or less, the position and the stability of the periodic orbits. It is shown that the mass of the nucleus, the mass of the disk, the halo flattening parameter, the scale length of the halo, the angular momentum, and the total orbital energy are the most influential quantities, while the effect of all other parameters is much weaker., Published in Baltic Astronomy journal

In this paper we use the two-dimensional (2D) version of a new analytical gravitational model in order to explore the orbital as well as the escape dynamics of the stars in a barred galaxy composed of a spherically symmetric central nucleus, a bar, a flat disk and a dark matter halo component. A thorough numerical investigation is conducted for distinguishing between bounded and escaping motion. Furthermore bounded orbits are further classified into non-escaping regular and trapped chaotic using the Smaller ALingment Index (SALI) method. Our aim is to determine the basins of escape through the two symmetrical escape channels around the Lagrange points $L_2$ and $L_3$ and also to relate them with the corresponding distribution of the escape rates of the orbits. We integrate initial conditions of orbits in several types of planes so as to obtain a more complete view of the overall orbital properties of the dynamical system. We also present evidence that the unstable manifolds which guide the orbits in and out the interior region are directly related with the formation of spiral and ring stellar structures observed in barred galaxies. In particular, we examine how the bar's semi-major axis determines the resulting morphologies. Our numerical simulations indicate that weak barred structures favour the formation of $R_1$ rings or $R_1'$ pseudo-rings, while strong bars on the other hand, give rise to $R_1R_2$ and open spiral morphologies. Our results are compared with earlier related work. The escape dynamics and the properties of the manifolds of the full three-dimensional (3D) galactic system will be given in an accompanying paper., Comment: Published in Monthly Notices of the Royal Astronomical Society (MNRAS) journal

The escape dynamics in an analytical gravitational model which describes the motion of stars in a binary system of interacting dwarf spheroidal galaxies is investigated in detail. We conduct a numerical analysis distinguishing between regular and chaotic orbits as well as between trapped and escaping orbits, considering only unbounded motion for several energy levels. In order to distinguish safely and with certainty between ordered and chaotic motion, we apply the Smaller ALingment Index (SALI) method. It is of particular interest to locate the escape basins through the openings around the collinear Lagrangian points $L_1$ and $L_2$ and relate them with the corresponding spatial distribution of the escape times of the orbits. Our exploration takes place both in the configuration $(x,y)$ and in the phase $(x,\dot{x})$ space in order to elucidate the escape process as well as the overall orbital properties of the galactic system. Our numerical analysis reveals the strong dependence of the properties of the considered escape basins with the total orbital energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. It was also observed, that for energy levels close to the critical escape energy the escape rates of orbits are large, while for much higher values of energy most of the orbits have low escape periods or they escape immediately to infinity. We hope our outcomes to be useful for a further understanding of the escape mechanism in binary galaxy models., Comment: Published in New Astronomy journal. arXiv admin note: previous papers with related context: arXiv:1604.04613, arXiv:1505.03968, arXiv:1508.05198, arXiv:1411.4864, arXiv:1511.04908, arXiv:1505.04185

The regular or chaotic dynamics of an analytical realistic three dimensional model composed of a spherically symmetric central nucleus, a bar and a flat disk is investigated. For describing the properties of the bar we introduce a new simple dynamical model and we explore the influence on the character of orbits of all the involved parameters of it, such as the mass and the scale length of the bar, the major semi-axis and the angular velocity of the bar as well as the energy. Regions of phase space with ordered and chaotic motion are identified in dependence on these parameters and for breaking the rotational symmetry. First we study in detail the dynamics in the invariant plane $z = p_z = 0$ using the Poincar\'e map as a basic tool and then we study the full 3 dimensional case using the SALI method as principal tool for distinguishing between order and chaos. We also present strong evidence obtained through the numerical simulations that our new bar model can realistically describe the formation and the evolution of the observed twin spiral structure in barred galaxies., Comment: Published in Publications of the Astronomical Society of Australia (PASA) journal