Effective power-law dependence of Lyapunov exponents on the central mass in galaxies

Using both numerical and analytical approaches, we demonstrate the existence of an effective power-law relation $L\propto m^p$ between the mean Lyapunov exponent $L$ of stellar orbits chaotically scattered by a supermassive black hole in the center of a galaxy and the mass parameter $m$, i.e. ratio of the mass of the black hole over the mass of the galaxy. The exponent $p$ is found numerically to obtain values in the range $p \approx 0.3$--$0.5$. 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 black hole's sphere of influence. We thus predict $p=2/3-q$ with $q\approx 0.1$--$0.2$. 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 black hole affects mainly populations of thick tube-like orbits surrounding some low-order branches of the $x_1$ family of periodic orbits, as well as its bifurcations at low-order resonances, mainly the Inner Lindbland 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.Comment: 27 pages, 19 figure

NGC 1300 Dynamics: III. Orbital analysis

We present the orbital analysis of four response models, that succeed in reproducing morphological features of NGC 1300. Two of them assume a planar (2D) geometry with $\Omega_p$=22 and 16 \ksk respectively. The two others assume a cylindrical (thick) disc and rotate with the same pattern speeds as the 2D models. These response models reproduce most successfully main morphological features of NGC 1300 among a large number of models, as became evident in a previous study. Our main result is the discovery of three new dynamical mechanisms that can support structures in a barred-spiral grand design system. These mechanisms are presented in characteristic cases, where these dynamical phenomena take place. They refer firstly to the support of a strong bar, of ansae type, almost solely by chaotic orbits, then to the support of spirals by chaotic orbits that for a certain number of pat tern revolutions follow an n:1 (n=7,8) morphology, and finally to the support of spiral arms by a combination of orbits trapped around L$_{4,5}$ and sticky chaotic orbits with the same Jacobi constant. We have encountered these dynamical phenomena in a large fraction of the cases we studied as we varied the parameters of our general models, without forcing in some way their appearance. This suggests that they could be responsible for the observed morphologies of many barred-spiral galaxies. Comparing our response models among themselves we find that the NGC 130 0 morphology is best described by a thick disc model for the bar region and a 2D disc model for the spirals, with both components rotating with the same pattern speed $\Omega_p$=16 \ksk !. In such a case, the whole structure is included inside the corotation of the system. The bar is supported mainly by regular orbits, while the spirals are supported by chaotic orbits.Comment: 18 pages, 32 figures, accepted for publication in MNRA

Particle Acceleration in Dissipative Pulsar Magnetospheres

Pulsar magnetospheres represent unipolar inductor-type electrical circuits at which an EM potential across the polar cap (due to the rotation of their magnetic field) drives currents that run in and out of the polar cap and close at infinity. An estimate ofthe magnitude of this current can be obtained by dividing the potential induced across the polar cap V approx = B(sub O) R(sub O)(Omega R(sub O)/c)(exp 2) by the impedance of free space Z approx eq 4 pi/c; the resulting polar cap current density is close to $n {GJ} c$ where $n_{GJ}$ is the Goldreich-Julian (GJ) charge density. This argument suggests that even at current densities close to the GJ one, pulsar magnetospheres have a significant component of electric field $E_{parallel}$, parallel to the magnetic field, a condition necessary for particle acceleration and the production of radiation. We present the magnetic and electric field structures as well as the currents, charge densities, spin down rates and potential drops along the magnetic field lines of pulsar magnetospheres which do not obey the ideal MHD condition $E cdot B = 0$. By relating the current density along the poloidal field lines to the parallel electric field via a kind of Ohm's law $J = sigma E_{parallel}$ we study the structure of these magnetospheres as a function of the conductivity $sigma$. We find that for $sigma gg OmegaS the solution tends to the (ideal) Force-Free one and to the Vacuum one for$sigma 11 OmegaS. Finally, we present dissipative magnetospheric solutions with spatially variable $sigma$ that supports various microphysical properties and are compatible with the observations

NGC 1300 Dynamics: II. The response models

We study the stellar response in a spectrum of potentials describing the barred spiral galaxy NGC 1300. These potentials have been presented in a previous paper and correspond to three different assumptions as regards the geometry of the galaxy. For each potential we consider a wide range of $\Omega_p$ pattern speed values. Our goal is to discover the geometries and the $\Omega_p$ supporting specific morphological features of NGC 1300. For this purpose we use the method of response models. In order to compare the images of NGC 1300 with the density maps of our models, we define a new index which is a generalization of the Hausdorff distance. This index helps us to find out quantitatively which cases reproduce specific features of NGC 1300 in an objective way. Furthermore, we construct alternative models following a Schwarzschild type technique. By this method we vary the weights of the various energy levels, and thus the orbital contribution of each energy, in order to minimize the differences between the response density and that deduced from the surface density of the galaxy, under certain assumptions. We find that the models corresponding to $\Omega_p\approx16$\ksk and $\Omega_p\approx22$\ksk are able to reproduce efficiently certain morphological features of NGC 1300, with each one having its advantages and drawbacks.Comment: 13 pages, 10 figures, accepted for publication in MNRA

Chaotic motion and spiral structure in self-consistent models of rotating galaxies

Dissipationless N-body models of rotating galaxies, iso-energetic to a non-rotating model, are examined as regards the mass in regular and in chaotic motion. The values of their spin parameters $\lambda$ are near the value $\lambda=0.22$ of our Galaxy. We obtain the distinction between the sets of particles moving in regular and in chaotic orbits and we show that the spatial distribution of these two sets of particles is much different. The rotating models are characterized by larger fractions of mass in chaotic motion ($\thickapprox 65$) compared with the fraction of mass in chaotic motion in the non-rotating iso-energetic model ($\thickapprox 32$). Furthermore, the Lyapunov numbers of the chaotic orbits in the rotating models become by about one order of magnitude larger than in the non-rotating model. Chaotic orbits are concentrated preferably in values of the Jacobi integral around the value of the effective potential at the corotation radius. We find that density waves form a central rotating bar embedded in a thin and a thick disc with exponential surface density profile. A surprising new result is that long living spiral arms are exited on the disc, composed almost completely by chaotic orbits. The bar excites an $m=2$ mode of spiral waves on the surface density of the disc, emanating from the corotation radius. These spiral waves are deformed, fade, or disappear temporarily, but they grow again re-forming a well developed spiral pattern. Spiral arms are discernible up to 20 or 30 rotations of the bar (lasting for about a Hubble time).Comment: 30 pages, 17 figures (low resolution). Revised version. Accepted for publication in MNRAS. For high resolution figures please send email to [email protected]

Modeling magnetic disk wind state transitions in black hole X-Ray binaries

We analyze three prototypical black hole X-ray binaries, 4U 1630-472, GRO J1655-40, and H1743-322, in an effort to systematically understand the intrinsic state transition of the observed accretion disk winds between wind-on and wind-off states by utilizing state-of-the-art Chandra/HETGS archival data from multi-epoch observations. We apply our magnetically driven wind models in the context of magnetohydrodynamic (MHD) calculations to constrain (1) their global density slope (p), (2) their density (n (17)) at the foot point of the innermost launching radius, and (3) the abundances of heavier elements (A (Fe,S,Si)). Incorporating the MHD winds into xstar photoionization calculations in a self-consistent manner, we create a library of synthetic absorption spectra given the observed X-ray continua. Our analysis clearly indicates a characteristic bimodal transition of multi-ion X-ray winds; i.e., the wind density gradient is found to steepen (from p similar to 1.2-1.4 to similar to 1.4-1.5) while its density normalization declines as the source transitions from the wind-on to the wind-off state. The model implies that the ionized wind remains physically present even in the wind-off state, despite its apparent absence in the observed spectra. Supersolar abundances for heavier elements are also favored. Our global multi-ion wind models, taking into account soft X-ray ions as well as Fe K absorbers, show that the internal wind condition plays an important role in wind transitions besides photoionization changes. Simulated XRISM/Resolve and Athena/X-IFU spectra are presented to demonstrate a high fidelity of the multi-ion wind model for a better understanding of these powerful ionized winds in the coming decades

Nodal points and the transition from ordered to chaotic Bohmian trajectories

We explore the transition from order to chaos for the Bohmian trajectories of a simple quantum system corresponding to the superposition of three stationary states in a 2D harmonic well with incommensurable frequencies. We study in particular the role of nodal points in the transition to chaos. Our main findings are: a) A proof of the existence of bounded domains in configuration space which are devoid of nodal points, b) An analytical construction of formal series representing regular orbits in the central domain as well as a numerical investigation of its limits of applicability. c) A detailed exploration of the phase-space structure near the nodal point. In this exploration we use an adiabatic approximation and we draw the flow chart in a moving frame of reference centered at the nodal point. We demonstrate the existence of a saddle point (called X-point) in the vicinity of the nodal point which plays a key role in the manifestation of exponential sensitivity of the orbits. One of the invariant manifolds of the X-point continues as a spiral terminating at the nodal point. We find cases of Hopf bifurcation at the nodal point and explore the associated phase space structure of the nodal point - X-point complex. We finally demonstrate the mechanism by which this complex generates chaos. Numerical examples of this mechanism are given for particular chaotic orbits, and a comparison is made with previous related works in the literature.Comment: 32 pages, 13 figures, Accepted for publication in Journal of Physics

Nonlinear force-free reconstruction of the global solar magnetic field: methodology

We present a novel numerical method that allows the calculation of nonlinear force-free magnetostatic solutions above a boundary surface on which only the distribution of the normal magnetic field component is given. The method relies on the theory of force-free electrodynamics and applies directly to the reconstruction of the solar coronal magnetic field for a given distribution of the photospheric radial field component. The method works as follows: we start with any initial magnetostatic global field configuration (e.g. zero, dipole), and along the boundary surface we create an evolving distribution of tangential (horizontal) electric fields that, via Faraday's equation, give rise to a respective normal field distribution approaching asymptotically the target distribution. At the same time, these electric fields are used as boundary condition to numerically evolve the resulting electromagnetic field above the boundary surface, modelled as a thin ideal plasma with non-reflecting, perfectly absorbing outer boundaries. The simulation relaxes to a nonlinear force-free configuration that satisfies the given normal field distribution on the boundary. This is different from existing methods relying on a fixed boundary condition - the boundary evolves toward the a priori given one, at the same time evolving the three-dimensional field solution above it. Moreover, this is the first time a nonlinear force-free solution is reached by using only the normal field component on the boundary. This solution is not unique, but depends on the initial magnetic field configuration and on the evolutionary course along the boundary surface. To our knowledge, this is the first time that the formalism of force-free electrodynamics, used very successfully in other astrophysical contexts, is applied to the global solar magnetic field.Comment: 18 pages, 5 figures, Solar Physic

Invariant manifolds and the response of spiral arms in barred galaxies

The unstable invariant manifolds of the short-period family of periodic orbits around the unstable Lagrangian points $L_1$ and $L_2$ of a barred galaxy define loci in the configuration space which take the form of a trailing spiral pattern. In the present paper we investigate this association in the case of the self-consistent models of Kaufmann & Contopoulos (1996) which provide an approximation of real barred-spiral galaxies. We also examine the relation of `response' models of barred-spiral galaxies with the theory of the invariant manifolds. Our main results are the following: The invariant manifolds yield the correct form of the imposed spiral pattern provided that their calculation is done with the spiral potential term turned on. We provide a theoretical model explaining the form of the invariant manifolds that supports the spiral structure. The azimuthal displacement of the Lagrangian points with respect to the bar's major axis is a crucial parameter in this modeling. When this is taken into account, the manifolds necessarily develop in a spiral-like domain of the configuration space, delimited from below by the boundary of a banana-like non-permitted domain, and from above either by rotational KAM tori or by cantori forming a stickiness zone. We construct `spiral response' models on the basis of the theory of the invariant manifolds and examine the connection of the latter to the `response' models (Patsis 2006) used to fit real barred-spiral galaxies, explaining how are the manifolds related to a number of morphological features seen in such models.Comment: 16 Page