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Effective Power-Law Dependence of Lyapunov Exponents on the Central Mass in GalaxiesUsing both numerical and analytical approaches, we demonstrate the existence of an effective power-law relation L alpha m(sup p) between themean Lyapunov exponent L of stellar orbits chaotically scattered by a supermassive black hole (BH) in the centre of a galaxy and the mass parameter m, i.e. ratio of the mass of the BH over the mass of the galaxy. The exponent p is found numerically to obtain values in the range p approximately equals 0.3-0.5. We propose a theoretical interpretation of these exponents, based on estimates of local 'stretching numbers', i.e. local Lyapunov exponents at successive transits of the orbits through the BH's sphere of influence. We thus predict p = 2/3 − q with q approximately equaling 0.1-0.2. Our basic model refers to elliptical galaxy models with a central core. However, we find numerically that an effective power-law scaling of L with m holds also in models with central cusp, beyond a mass scale up to which chaos is dominated by the influence of the cusp itself. We finally show numerically that an analogous law exists also in disc galaxies with rotating bars. In the latter case, chaotic scattering by the BH affects mainly populations of thick tube-like orbits surrounding some low-order branches of the x(sub 1) family of periodic orbits, as well as its bifurcations at low-order resonances, mainly the inner Lindblad resonance and the 4/1 resonance. Implications of the correlations between L and m to determining the rate of secular evolution of galaxies are discussed.
Document ID
Document Type
Reprint (Version printed in journal)
External Source(s)
Delis, N. (Athens Univ. Greece)
Efthymiopoulos, C. (Academy of Athens Greece)
Kalapotharakos, C. (Maryland Univ. College Park, MD, United States)
Date Acquired
April 20, 2016
Publication Date
March 4, 2015
Publication Information
Publication: Monthly Notices of the Royal Astronomical Society
Volume: 448
Issue: 3
Subject Category
Report/Patent Number
Funding Number(s)
Distribution Limits
chaos aEuro" galaxies:
kinematics and dynamics
kinematics and dynamics