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Register a new userDynamics of Cuspy Triaxial Galaxies with a Supermassive Black Hole
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This talk provides a progress report on an extended collaboration which has aimed to address two basic questions, namely: Should one expect to see cuspy, triaxial galaxies in nature? And can one construct realistic cuspy, triaxial equilibrium models that are robust? Three technical results are described: (1) Unperturbed chaotic orbits in cuspy triaxial potentials can be extraordinarily sticky, much more so than orbits in many other three-dimensional potentials. (2) Even very weak perturbations can be important by drastically reducing, albeit not completely eliminating, this stickiness. (3) A simple toy model facilitates a simple understanding of why black holes and cusps can serve as an effective source of chaos. These results suggest that, when constructing models of galaxies using Schwarzschilds method or any analogue thereof, astronomers would be well advised to use orbital building blocks that have been perturbed by `noise or other weak irregularities, since such building blocks are likely to be more nearly time-independent than orbits evolved in the absence of all perturbations.
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This paper summarises an investigation of chaos in a toy potential which mimics much of the behaviour observed for the more realistic triaxial generalisations of the Dehnen potentials, which have been used to model cuspy triaxial galaxies both with and without a supermassive black hole. The potential is the sum of an anisotropic harmonic oscillator potential, V_o=(1/2)*(a^{2}x^{2}+b^{2}y^{2}+c^{2}z^{2}), and a spherical Plummer potential, V_o=-M_{BH}/(r^{2}+e^{2})^{1/2} with r^{2}=x^{2}+y^{2}+z^{2}. Attention focuses on three issues related to the properties of ensembles of chaotic orbits which impact on chaotic mixing and the possibility of constructing self-consistent equilibria: (1) What fraction of the orbits are chaotic? (2) How sensitive are the chaotic orbits, i.e., how large are their largest (short time) Lyapunov exponents? (3) To what extent is the motion of chaotic orbits impeded by Arnold webs, i.e.,, how `sticky are the chaotic orbits? These questions are explored as functions of the axis ratio a:b:c, black hole mass M_BH, softening length e, and energy E with the aims of understanding how the manifestations of chaos depend on the shape of the system and why the black hole generates chaos. The simplicity of the model makes it amenable to a perturbative analysis. That it mimics the behaviour of more complicated potentials suggests that much of this behaviour should be generic.
We have constructed realistic, self-consistent models of triaxial elliptical galaxies embedded in triaxial dark matter halos. Self-consistent solutions by means of the standard orbital superposition technique introduced by Schwarzschild were found in each of the three cases studied. Chaotic orbits were found to be important in all of the models, and their presence was shown to imply a possible slow evolution of the shapes of the halos. The equilibrium velocity distribution is reproduced by a Lorentzian function better than by a Gaussian. Our results demonstrate for the first time that triaxial dark matter halos can co-exist with triaxial galaxies.
Cuspy triaxial potentials admit a large number of chaotic orbits, which moreover exhibit extreme stickiness that makes the process of chaotic mixing surprisingly inefficient. Environmental effects, modeled as noise and/or periodic driving, help accelerate phase space transport but probably not as much as in simpler potentials. This could mean that cuspy triaxial ellipticals cannot exist as time-independent systems.
Cuspy shadow was first reported for hairy rotating black holes, whose metrics deviate significantly from the Kerr one. The non-smooth edge of the shadow is attributed to a transition between different branches of unstable but bounded orbits, known as the fundamental photon orbits, which end up at the light rings. In searching for a minimal theoretical setup to reproduce such a salient feature, in this work, we devise a toy model with axisymmetry, a slowly rotating Kerr black hole enveloped by a thin slowly rotating dark matter shell. Despite its simplicity, we show rich structures regarding fundamental photon orbits explicitly in such a system. We observe two disconnected branches of unstable spherical photon orbits, and the jump between them gives rise to a pair of cusps in the resultant black hole shadow. Besides the cuspy shadow, we explore other intriguing phenomena when the Maxwell construction cannot be established. We find that it is possible to have an incomplete arc of Einstein rings and a fractured shadow. The potential astrophysical significance of the corresponding findings is addressed.
In galactic nuclei, the gravitational potential is dominated by the central supermassive black hole, so stars follow quasi-Keplerian orbits. These orbits are distorted by gravitational forces from other stars, leading to long-term orbital relaxation. The direct numerical study of these processes is challenging because the fast orbital motion imposed by the central black hole requires very small timesteps. Within the secular approximation of smearing out stars along their underlying Keplerian orbits, a multipole expansion of the pairwise interaction between the stars yields an efficient numerical code to investigate the long-term evolution of their orbital parameters. These new simulations precisely recover the diffusion coefficients of stellar eccentricities obtained through analytical calculations of the secular dynamics. The computational complexity of the present method scales linearly with the total number of stars, so it should prove useful to study the long-term evolution of self-gravitating systems dominated by a central mass.
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