Trapped and Escaping Orbits in an Axially Symmetric Galactic-Type Potential

In the present article, we investigate the behavior of orbits in a time-independent axially symmetric galactic-type potential. This dynamical model can be considered to describe the motion in the central parts of a galaxy, for values of energies larger than the energy of escape. We use the classical surface-of-section method in order to visualize and interpret the structure of the phase space of the dynamical system. Moreover, the Lyapunov characteristic exponent is used in order to make an estimation of the degree of chaoticity of the orbits in our galactic model. Our numerical calculations suggest that in this galactic-type potential there are two kinds of orbits: (i) escaping orbits and (ii) trapped orbits, which do not escape at all. Furthermore, a large number of orbits of the dynamical system display chaotic motion. Among the chaotic orbits, there are orbits that escape quickly and also orbits that remain trapped for vast time intervals. When the value of a test particle's energy slightly exceeds the energy of escape, the number of trapped regular orbits increases as the value of the angular momentum increases. Therefore, the extent of the chaotic regions observed in the phase plane decreases as the energy value increases. Moreover, we calculate the average value of the escape period of chaotic orbits and try to correlate it with the value of the energy and also with the maximum value of the z component of the orbits. In addition, we find that the value of the Lyapunov characteristic exponent corresponding to each chaotic region for different values of energy increases exponentially as the energy increases. Some theoretical arguments are presented in order to support the numerically obtained outcomes.