Mass components in ordered and in chaotic motion in galactic N-body models

Abstract

Two self-consistent (N-body) non-rotating equilibrium models of elliptical galaxies with smooth central density profiles (called 'Q' and 'C' models) are constructed, starting from quiet and clumpy cosmological initial conditions, respectively. Both models are triaxial. The Q model has an E7 maximum ellipticity in the inner parts and tends to E6 or E5 maximum ellipticity in the outer parts. The C model has a maximum ellipticity E4 in the inner parts and tends to an E2 or E1 in the outer parts.For each model, we identify the particles moving in chaotic orbits with the Lyapunov number exceeding a particular threshold (namely, 10 −2.8 , in units of the inverse radial periods of the particular orbits). At energy levels in the deepest 30 per cent of the potential well, no chaotic orbits were detected in the above limit of chaoticity. In the Q model, the detected chaotic part is 32 per cent of the total mass. This part has a nearly spherical distribution. It imposes limitations on the maximum ellipticity of the system, in spite of the fact that only a part of less than about 8 per cent of the total mass moves in chaotic orbits and is able to develop chaotic diffusion within a Hubble time. In the C model, the detected chaotic part is about 26 per cent of the total mass, but only less than 2 per cent can develop chaotic diffusion within a Hubble time.These chaotic components produce surface density profiles flatter than the profiles of the rest of the mass, particularly in the Q model. The two profiles intersect at a given distance, where the overall profile forms an observable hump, especially if the surface density profiles are taken along the shortest axis of the projection.

Adjusting chaotic indicators to curved spacetimes

“…The Generalized Alignment Index [11] is a generalization of the Smaller Alignment Index (SALI) [10] (also called Alignment Index [35]). GALI differs from the indicators discussed above because it does not depend on the rate by which a deviation vector grows, but on whether two or more deviation vectors with different initial directions will get aligned or not.…”

Special Features of Galactic Dynamics

“…The distribution of chaotic orbits is almost spherical while the distribution of regular orbits is strongly elongated along the long axis of the system. (after Voglis et al 2002).…”

“…Such differences can be quantified by the introduction of appropriate measures of the inverse Lyapunov number, i.e., the Lyapunon time of orbits (e.g. Voglis et al 2002).…”

“…A extended study of global dynamics in N-Body systems was made by Voglis et al (2002), Kalapotharakos et al (2004) and Kalapotharakos and Voglis (2005). This is a study of N-Body systems in equilibrium resulting from the cosmological simulations described in subsection 2.5.…”

“…The question then is what is a proper normalization of Lyapunov times (or Lyapunov exponents) that provides a fair measure of the 'chaoticity' of an orbit. In order to address this question, Voglis et al (2002) introduced a new type of calculation based on the combination of two methods: a) the 'specific time Lyapunov number' L j , normalized with respect to the orbit's inverse of the period, and b) the Alignment Index (AI). The Alignment Index is a numerical method based on certain properties of the time evolution of deviation vectors (Voglis et al, 1998, Skokos, 2001.…”