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Register a new userDiscreteness effects, $N-$body chaos and the onset of radial-orbit instability
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Added by
Pierfrancesco Di Cintio
Publication date
2019
fields
Physics
and research's language is
English
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We study the stability of a family of spherical equilibrium models of self-gravitating systems, the so-called $gamma-$models with Osipkov-Merritt velocity anisotropy, by means of $N-$body simulations. In particular, we analyze the effect of self-consistent $N-$body chaos on the onset of radial-orbit instability (ROI). We find that degree of chaoticity of the system associated to its largest Lyapunov exponent $Lambda_{rm max}$ has no appreciable relation with the stability of the model for fixed density profile and different values of radial velocity anisotropy. However, by studying the distribution of the Lyapunov exponents $lambda_{rm m}$ of the individual particles in the single-particle phase space, we find that more anisotropic systems have a larger fraction of orbits with larger $lambda_{rm m}$.
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We revisit the r^{o}le of discreteness and chaos in the dynamics of self-gravitating systems by means of $N$-body simulations with active and frozen potentials, starting from spherically symmetric stationary states and considering the orbits of single particles in a frozen $N$-body potential as well as the orbits of the system in the full $6N$-dimensional phase space. We also consider the intermediate case where a test particle moves in the field generated by $N$ non-interacting particles, which in turn move in a static smooth potential. We investigate the dependence on $N$ and on the softening length of the largest Lyapunov exponent both of single particle orbits and of the full $N$-body system. For single orbits we also study the dependence on the angular momentum and on the energy. Our results confirm the expectation that orbital properties of single orbits in finite-$N$ systems approach those of orbits in smooth potentials in the continuum limit $N to infty$ and that the largest Lyapunov exponent of the full $N$-body system does decrease with $N$, for sufficiently large systems with finite softening length. However, single orbits in frozen models and active self-consistent models have different largest Lyapunov exponents and the $N$-dependence of the values in non-trivial, so that the use of frozen $N$-body potentials to gain information on large-$N$ systems or on the continuum limit may be misleading in certain cases.
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Using direct $N$-body simulations of self-gravitating systems we study the dependence of dynamical chaos on the system size $N$. We find that the $N$-body chaos quantified in terms of the largest Lyapunov exponent $Lambda_{rm max}$ decreases with $N$. The values of its inverse (the so-called Lyapunov time $t_lambda$) are found to be smaller than the two-body collisional relaxation time but larger than the typical violent relaxation time, thus suggesting the existence of another collective time scale connected to many-body chaos.
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