Revealing the dynamical behavior of stars in a prolate or oblate elliptical galaxy

We consider the motion of a test particle (star) on the meridian plane (R,z) of an elliptical galaxy with an additional central nucleus component. The shape of the elliptical galaxy can be either prolate or oblate, or even spherical by simply changing the numerical value of the flattening parameter entering the equation of the potential of the elliptical galaxy. By means of numerical integration of both the motion equations and the equations of variations of the system, we conduct a systematic orbit classification of the starting conditions of the test particle, thus revealing how the flattening parameter affects its orbital dynamics. Our simulations suggest that in the scenario where the elliptical galaxy has a prolate shape the motion of a star is much more complicated with respect to the case where we have a usual oblate elliptical galaxy. Additionally, we also investigate how the total orbital energy of the test particle affects its dynamics in relation to the flattening parameter of the elliptical galaxy.