A Note on the Toda Criterion for Interacting Dipole-Quadrupole Vibrations

The Toda criterion of the Gaussian curvature is applied to calculate analytically the transition energy from regular to chaotic motion in a schematic model describing the interaction between collective dipole and quadrupole modes in atomic nuclei.Comment: Latex, 9 pages, 2 figures (available upon request), to be published in Modern Physics Letters

Quantum to classical transition in a system with a mixed classical dynamics

We study how decoherence rules the quantum-classical transition of the Kicked Harmonic Oscillator (KHO). When the amplitude of the kick is changed the system presents a classical dynamics that range from regular to a strong chaotic behavior. We show that for regular and mixed classical dynamics, and in the presence of noise, the distance between the classical and the quantum phase space distributions is proportional to a single parameter $\chi\equiv K\hbar_{\rm eff}^2/4D^{3/2}$ which relates the effective Planck constant $\hbar_{\rm eff}$, the kick amplitude $K$ and the diffusion constant $D$. This is valid when $\chi < 1$, a case that is always attainable in the semiclassical regime independently of the value of the strength of noise given by $D$. Our results extend a recent study performed in the chaotic regime.Comment: 10 pages, 7 figure

On the Origin of Cusps in Stellar Systems

An origin is sought for the ubiquity of cusps, both in computer simulations of halo formation in hierarchical clustering cosmogonies and in observations of galactic nuclei by the Hubble Space Telescope (HST). The encounters of merging clumps that built the galaxies can be described by the collisional Boltzmann equation. Using insights gained by studying the simpler Fokker-Planck equation, we show that there is a steady-state, self-consistent, cusped solution of the collisional Boltzmann equation corresponding to $\rho \sim r^{-4/3}$. This equilibrium is both stable and an attractor. It is the natural end-point of the diffusive encounters of an ensemble of equal mass clumps. The introduction of a mass spectrum weakens the mass density cusp. The spike in the luminosity density can be accentuated or softened, depending on the form of the mass-luminosity relation. Possible applications to the cusped nuclei of early-type galaxies are discussed.Comment: Latex, 14 pages, Needs aasms4.sty. The Astrophysical Journal (Letters), in pres

Partial suppression of the radial orbit instability in stellar systems

It is well known that the simple criterion proposed originally by Polyachenko and Shukhman (1981) for the onset of the radial orbit instability, although being generally a useful tool, faces significant exceptions both on the side of mildly anisotropic systems (with some that can be proved to be unstable) and on the side of strongly anisotropic models (with some that can be shown to be stable). In this paper we address two issues: Are there processes of collisionless collapse that can lead to equilibria of the exceptional type? What is the intrinsic structural property that is responsible for the sometimes noted exceptional stability behavior? To clarify these issues, we have performed a series of simulations of collisionless collapse that start from homogeneous, highly symmetrized, cold initial conditions and, because of such special conditions, are characterized by very little mixing. For these runs, the end-states can be associated with large values of the global pressure anisotropy parameter up to 2K_r/K_T \approx 2.75. The highly anisotropic equilibrium states thus constructed show no significant traces of radial anisotropy in their central region, with a very sharp transition to a radially anisotropic envelope occurring well inside the half-mass radius (around 0.2 r_M). To check whether the existence of such almost perfectly isotropic "nucleus" might be responsible for the apparent suppression of the radial orbit instability, we could not resort to equilibrium models with the above characteristics and with analytically available distribution function; instead, we studied and confirmed the stability of configurations with those characteristics by initializing N-body approximate equilibria (with given density and pressure anisotropy profiles) with the help of the Jeans equations.Comment: 26 pages, 9 figures, accepted for publication in The Astrophysical Journa

Numerical stability of a family of Osipkov-Merrit models

We have investigated the stability of a set of non-rotating anisotropic spherical models with a phase-space distribution function of the Osipkov-Merritt type. The velocity distribution in these models is isotropic near the center and becomes radially anisotropic at large radii. They are special members of the family studied by Dehnen and Tremaine et al. where the mass density has a power-law cusp $\rho\propto r^{-\gamma}$ at small radii and decays as $\rho\propto r^{-4}$ at large radii. The radial-orbit instability of models with $\gamma$ = 0, 1/2, 1, 3/2, and 2, was studied using an N-body code written by one of us and based on the `self-consistent field' method developed by Hernquist and Ostriker. These simulations have allowed us to delineate a boundary in the $(\gamma,r_{a})$-plane that separates the stable from the unstable models. This boundary is given by $2T_{r}/T_{t} = 2.31 \pm 0.27$, for the ratio of the total radial to tangential kinetic energy. We also found that the stability criterion $df/dQ\le 0$, recently raised by Hjorth, gives lower values compared with our numerical results.Comment: AASTEX, 22 pages, 11 figures, Figs. 5 available from author. Accepted for publication in Astrophysical Journa

Mean Field Theory of Spherical Gravitating Systems

Important gaps remain in our understanding of the thermodynamics and statistical physics of self-gravitating systems. Using mean field theory, here we investigate the equilibrium properties of several spherically symmetric model systems confined in a finite domain consisting of either point masses, or rotating mass shells of different dimension. We establish a direct connection between the spherically symmetric equilibrium states of a self-gravitating point mass system and a shell model of dimension 3. We construct the equilibrium density functions by maximizing the entropy subject to the usual constraints of normalization and energy, but we also take into account the constraint on the sum of the squares of the individual angular momenta, which is also an integral of motion for these symmetric systems. Two new statistical ensembles are introduced which incorporate the additional constraint. They are used to investigate the possible occurrence of a phase transition as the defining parameters for each ensemble are altered

Reduction and Realization in Toda and Volterra

We construct a new symplectic, bi-hamiltonian realization of the KM-system by reducing the corresponding one for the Toda lattice. The bi-hamiltonian pair is constructed using a reduction theorem of Fernandes and Vanhaecke. In this paper we also review the important work of Moser on the Toda and KM-systems.Comment: 17 page

On Universal Halos and the Radial Orbit Instability

The radial orbit instability drives dark matter halos toward a universal structure. This conclusion, first noted by Huss, Jain, and Steinmetz, is explored in detail through a series of numerical experiments involving the collapse of an isolated halo into the non-linear regime. The role played by the radial orbit instability in generating the density profile, shape, and orbit structure is carefully analyzed and, in all cases, the instability leads to universality independent of initial conditions. New insights into the underlying physics of the radial orbit instability are presented.Comment: 31 pages, 11 figures, submitted to the Astrophysical Journa

Classical Monopoles: Newton, NUT-space, gravomagnetic lensing and atomic spectra

Stimulated by a scholium in Newton's Principia we find some beautiful results in classical mechanics which can be interpreted in terms of the orbits in the field of a mass endowed with a gravomagnetic monopole. All the orbits lie on cones! When the cones are slit open and flattened the orbits are exactly the ellipses and hyperbolae that one would have obtained without the gravomagnetic monopole. The beauty and simplicity of these results has led us to explore the similar problems in Atomic Physics when the nuclei have an added Dirac magnetic monopole. These problems have been explored by others and we sketch the derivations and give details of the predicted spectrum of monopolar hydrogen. Finally we return to gravomagnetic monopoles in general relativity. We explain why NUT space has a non-spherical metric although NUT space itself is the spherical space-time of a mass with a gravomagnetic monopole. We demonstrate that all geodesics in NUT space lie on cones and use this result to study the gravitational lensing by bodies with gravomagnetic monopoles. We remark that just as electromagnetism would have to be extended beyond Maxwell's equations to allow for magnetic monopoles and their currents so general relativity would have to be extended to allow torsion for general distributions of gravomagnetic monopoles and their currents. Of course if monopoles were never discovered then it would be a triumph for both Maxwellian Electromagnetism and General Relativity as they stand!Comment: 39 pages, 9 figures and 2 tables available on request from the author

Stellar remnants in galactic nuclei: mass segregation

The study of how stars distribute themselves around a massive black hole (MBH) in the center of a galaxy is an important prerequisite for the understanding of many galactic-center processes. These include the observed overabundance of point X-ray sources at the Galactic center, the prediction of rates and characteristics of tidal disruptions of extended stars by the MBH and of inspirals of compact stars into the MBH, the latter being events of high importance for the future space borne gravitational wave interferometer LISA. In relatively small galactic nuclei, hosting MBHs with masses in the range 10^5-10^7 Msun, the single most important dynamical process is 2-body relaxation. It induces the formation of a steep density cusp around the MBH and strong mass segregation, as more massive stars lose energy to lighter ones and drift to the central regions. Using a spherical stellar dynamical Monte-Carlo code, we simulate the long-term relaxational evolution of galactic nucleus models with a spectrum of stellar masses. Our focus is the concentration of stellar black holes to the immediate vicinity of the MBH. We quantify this mass segregation for a variety of galactic nucleus models and discuss its astrophysical implications. Special attention is given to models developed to match the conditions in the Milky Way nucleus; we examine the presence of compact objects in connection to recent high-resolution X-ray observations.Comment: 28 pages, 24 figures, ApJ accepted. Small changes to follow referee's suggestion