Initial Conditions Of Orbits Research Articles

Orbital and escape dynamics in barred galaxies – I. The 2D system

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.

Interplay between Dark Matter and Galactic Structure in Disk and Oblate Elliptical Galaxies

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.

THREE-DIMENSIONAL p-q RESONANT ORBITS CLOSE TO SECOND SPECIES SOLUTIONS

The purpose of this paper is to study, for small values of µ, the three-dimensional p-q resonant orbits that are close to periodic second species solutions (SSS) of the restricted three-body problem. The work is based on an analytic study of the in- and out-maps. These maps are associated to follow, under the flow of the problem, initial conditions on a sphere of radius µ α around the small primary, and consider the images of those initial points on the same sphere. The out-map is associated to follow the flow forward in time and the in-map backwards. For both mappings we give analytical expressions in powers of the mass parameter. Once these expressions are obtained, we proceed to the study of the matching equations between both, obtaining initial conditions of orbits that will be 'periodic' with an error of the order µ 1−α ,f or someα ∈ (1/3, 1/2). Since, as µ → 0, the inner solution and the outer solution will collide with the small primary, these orbits will be close to SSS.