Mathematics – Differential Geometry
Scientific paper
Oct 2002
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2002phdt.........8t&link_type=abstract
Thesis (PhD). THE FLORIDA STATE UNIVERSITY, Source DAI-B 63/04, p. 1884, Oct 2002, 134 pages.
Mathematics
Differential Geometry
4
Scientific paper
We study the problem of building self-consistent models for triaxial elliptical galaxies both numerically and semi-analytically. Schwarzschild's orbit superposition method, a standard tool for numerical construction of self-consistent galaxy models, was implemented for the scale-free potentials with singular central density profiles, which bear direct relevance to recent observations. This comprehensive study spans virtually the entire range of physically possible models for galaxies with scale-free potentials and singular central density profiles. It provides the most exhaustive coverage of both density profiles and galactic shapes, thus enabling us to better understand the dependence of stability on these properties. The dependence of the method on its parameters, such as sampling of the phase space, coarse-graining of the configuration space, orbit classification, etc., has been discussed in great detail. This thorough analysis enables us to distinguish between the numerical artifacts and the intrinsic properties of the system before any conclusions about the dynamics are reached. We devise a method for building self-consistent galaxy models with separable potentials using a semi-analytic scheme which takes advantage of the potential's integrability. The scheme constructs the three-integral phase-space distribution functions for triaxial galaxy models with a gravitational potential of separable form, and an arbitrary triaxial luminous density distribution. It exploits our knowledge of analytic expressions for the distribution function of orbits without radial components ([1, 2]) and “thickens” them according to a prescribed radial dependence. The convergence of the scheme then necessarily implies self-consistency. In addition, we investigate the origin of chaos in galactic systems, as it seems to be a dominant feature in systems of two degrees of freedom or higher. Our numerical study of the scattering effects of the central galactic mass reveals a delicate interplay between stabilizing and scattering effects. We also use the tools of differential geometry to formulate the trajectories in a general Hamiltonian system with three degrees of freedom as geodesics on a Riemannian manifold equipped with a suitable metric. After such a formulation establishes the equivalence of two approaches, we exploit powerful tools of differential geometry to study the stability of individual orbits and orbital ensembles.
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