Orbital and escape dynamics in barred galaxies – II. The 3D system: exploring the role of the normally hyperbolic invariant manifolds

Abstract

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 Poincaré 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 1 R 2 rings and twin spiral formations, respectively. We also compare our numerical outcomes with earlier related work and with observational data.

Correlating the escape dynamics and the role of the normally hyperbolic invariant manifolds in a binary system of dwarf spheroidal galaxies

“…We study the dynamics of a test particle in the effective potential provided by this pair of galaxies. This system contains various index-1 saddles and one of them leads to a new kind of scenario which we did not yet see in the other examples studied so far (e.g., a barred galaxy [15] and a tidally limited star cluster [53]). An important step in the present paper is a preliminary attempt to classify the scenarios observed so far and to relate them with other properties of the dynamics around the respective saddle point.…”

“…Our system is a 3-dof system, therefore in the following we are interested in the case n = 3. In [15] we have given all the equations for the Lyapunov orbits and for the NHIM orbits in the quadratic approximation. These equations hold for any index-1 saddle of a 3-dof Hamiltonian system.…”

“…The present paper belongs to a series of papers with common scope (numerical investigation of the escape dynamics and the role of the NHIMs in Hamiltonian stellar systems) and therefore the structure as well as the numerical techniques are similar to those used in the previous papers (e.g., [15,53]). The article is organized as follows:…”

Geometry of escape and transition dynamics in the presence of dissipative and gyroscopic forces in two degree of freedom systems

“…The restricted three-body problem is a classic problem which has attracted a lot of attention for the study of escape [4][5][6][50][51][52][53][54][55][56][57][58][59][60][61][62][63]. Deeper understanding of the escape from a gravitational potential well due to multiple bodies can guide the design of trajectories for interplanetary space missions and also aid analysis of certain astronomical phenomena (e.g., galactic dynamics [64][65][66][67][68][69][70]). When dissipation is taken into account, the situation becomes more complicated, but becomes applicable to a larger number of situations, such as drag due to interplanetary dust [27,71], and engineering applications, such as low thrust [72] and solar sails [73,74].…”

Orbit classification and networks of periodic orbits in the planar circular restricted five-body problem

“…Over the years, several numerical methods and computational tools have been used for determining the character of orbits in the restricted three-body problem (RTBP) [52], as well as in other more complicated dynamical systems. These numerical techniques are mainly based on the computation of dynamical indicators, such as the Fast Lyapunov Indicator (FLI) (see e.g., [15,17,32]), the normally hyperbolic invariant manifolds (NHIMs), associated with the equilibrium points of the system (see e.g., [5,7,19,20,26,27,29,32,60]), or even the second species solutions (see e.g., [14]).…”