Showing papers by "Scott Tremaine published in 1999"

Axisymmetric, 3-Integral Models of Galaxies: A Massive Black Hole in NGC3379

Abstract: We fit axisymmetric 3-integral dynamical models to NGC3379 using the line-of-sight velocity distribution obtained from HST/FOS spectra of the galaxy center and ground-based long-slit spectroscopy along four position angles, with the light distribution constrained by WFPC2 and ground-based images. We have fitted models with inclinations from 29 (intrinsic galaxy type E5) to 90 degrees (intrinsic E1) and black hole masses from 0 to 1e9 M_solar. The best-fit black hole masses range from 6e7 to 2e8 M_solar, depending on inclination. The velocity ellipsoid of the best model is not consistent with either isotropy or a two-integral distribution function. Along the major axis, the velocity ellipsoid becomes tangential at the innermost bin, radial in the mid-range radii, and tangential again at the outermost bins. For the acceptable models, the radial to tangential dispersion in the mid-range radii ranges from 1.1 < sigma_r / sigma_t < 1.7. Compared with these 3-integral models, 2-integral isotropic models overestimate the black hole mass since they cannot provide adequate radial motion. However, the models presented in this paper still contain restrictive assumptions-namely assumptions of constant M/L and spheroidal symmetry-requiring yet more models to study black hole properties in complete generality.

A Class of Symplectic Integrators with Adaptive Time Step for Separable Hamiltonian Systems

Abstract: Symplectic integration algorithms are well suited for long-term integrations of Hamiltonian systems, because they preserve the geometric structure of the Hamiltonian flow. However, this desirable property is generally lost when adaptive time step control is added to a symplectic integrator. We describe an adaptive time step, symplectic integrator that can be used if the Hamiltonian is the sum of kinetic and potential energy components and the required time step depends only on the potential energy (e.g., test-particle integrations in fixed potentials). In particular, we describe an explicit, reversible, symplectic, leapfrog integrator for a test particle in a near-Keplerian potential; this integrator has a time step proportional to distance from the attracting mass and has the remarkable property of integrating orbits in an inverse-square force field with only "along-track" errors; i.e., the phase-space shape of a Keplerian orbit is reproduced exactly, but the orbital period is in error by O(N-2), where N is the number of steps per period.

A class of symplectic integrators with adaptive timestep for separable Hamiltonian systems

Abstract: Symplectic integration algorithms are well-suited for long-term integrations of Hamiltonian systems because they preserve the geometric structure of the Hamiltonian flow. However, this desirable property is generally lost when adaptive timestep control is added to a symplectic integrator. We describe an adaptive-timestep symplectic integrator that can be used if the Hamiltonian is the sum of kinetic and potential energy components and the required timestep depends only on the potential energy (e.g. test-particle integrations in fixed potentials). In particular, we describe an explicit, reversible, symplectic, leapfrog integrator for a test particle in a near-Keplerian potential; this integrator has timestep proportional to distance from the attracting mass and has the remarkable property of integrating orbits in an inverse-square force field with only "along-track" errors; i.e. the phase-space shape of a Keplerian orbit is reproduced exactly, but the orbital period is in error by O(1/N^2), where N is the number of steps per period.

The geometry of phase mixing

Abstract: Partially phase-mixed structures in galaxies occupy a complex surface of dimension D in six-dimensional phase space. The appearance of such structures to observers is determined by their projection into a subspace whose dimensionality K is determined by the number of observables (e.g. sky position, distance, radial velocity, etc.). We discuss the expected dimensionality of phase-space structures and suggest that the most prominent features in surveys with K ≥ D will be stable singularities (catastrophes). The simplest of these are the shells seen in the outer parts of elliptical galaxies.

Relaxation in stellar systems, and the shape and rotation of the inner dark halo

Abstract: Why do galactic bars rotate with high pattern speeds, when dynamical friction should rapidly couple the bar to the massive, slowly rotating dark halo? This long-standing paradox may be resolved by considering the dynamical interactions between the galactic disc and structures in the dark halo. Dynamical friction between small-scale halo structure and the disc spins up and flattens the inner halo, thereby quenching the dynamical friction exerted by the halo on the bar; at the same time the halo heats and thickens the inner disc, perhaps forming a rapidly rotating bulge. Two possible candidates for the required halo structures are massive black holes and tidal streamers from disrupted precursor haloes. More generally, gravitational scattering from phase-wrapped inhomogeneities represents a novel relaxation process in stellar systems, intermediate between violent relaxation and two-body relaxation, which can isotropize the distribution function at radii where two-body relaxation is not effective.