Determination of the dynamical structure of galaxies using optical spectra

Galaxy spectra are a rich source of kinematical information since the shapes of the absorption lines reflect the movement of stars along the line-of-sight. We present a technique with which to build directly a dynamical model for a galaxy by fitting model spectra, calculated from a dynamical model, to the observed galaxy spectra. Using synthetic spectra from a known galaxy model we demonstrate that this technique indeed recovers the essential dynamical characteristics of the galaxy model. Moreover, the method allows a statistically meaningful error analysis on the resulting dynamical quantities.     

Rates of tidal disruption of stars by massive central black holes

There is strong evidence for some kind of massive dark object in the centres of many galaxy bulges. The detection of flares from tidally disrupted stars could confirm that these objects are black holes (BHs). Here we present calculations of the stellar disruption rates in detailed dynamical models of real galaxies, taking into account the refilling of the loss cone of stars on disruptable orbits by two-body relaxation and tidal forces in non-spherical galaxies. The highest disruption rates (one star per 10⁴ yr) occur in faint ( L ≲10¹⁰ L_⊙) galaxies, which have steep central density cusps. More luminous galaxies are less dense and have much longer relaxation times and more massive BHs. Dwarf stars in such galaxies are swallowed whole by the BH and hence do not emit flares; giant stars could produce flares as often as every 10⁵ yr, although the rate depends sensitively on the shape of the stellar distribution function. We discuss the possibility of detecting disruption flares in current supernova searches. The total mass of stars consumed over the lifetime of the galaxy is of the order of 10⁶ M_⊙, independent of galaxy luminosity; thus, disrupted stars may contribute significantly to the present BH mass in galaxies fainter than ∼10⁹ L_⊙.     

Mass profiles and anisotropies of early-type galaxies

We discuss the problem of using stellar kinematics of early-type galaxies to constrain the orbital anisotropies and radial mass profiles of galaxies. We demonstrate that compressing the light distribution of a galaxy along the line of sight produces approximately the same signature in the line-of-sight velocity profiles as radial anisotropy. In particular, fitting spherically symmetric dynamical models to apparently round, isotropic face-on flattened galaxies leads to a spurious bias towards radial orbits in the models, especially if the galaxy has a weak face-on stellar disc. Such face-on stellar discs could plausibly be the cause of the radial anisotropy found in spherical models of intermediate luminosity ellipticals such as NGC 2434, 3379 and 6703.
In the light of this result, we use simple dynamical models to constrain the outer mass profiles of a sample of 18 round, early-type galaxies. The galaxies follow a Tully–Fisher relation parallel to that for spiral galaxies, but fainter by at least 0.8 mag ( I -band) for a given mass. The most luminous galaxies show clear evidence for the presence of a massive dark halo, but the case for dark haloes in fainter galaxies is more ambiguous. We discuss the observations that would be required to resolve this ambiguity.     

Stellar dynamics in razor-thin discs with massive nuclear black holes

The bifurcations of orbit-averaged dynamics are studied in a class of razor-thin discs with central black holes. The model used here consists of a perturbed harmonic oscillator Hamiltonian augmented with a GM r potential. Through a sequence of conformal and canonical transformations, we reduce the phase-space flows of the system to a set of non-linear differential equations on a sphere. Based on the critical points of the averaged system, we classify orbit families and reveal the existence of six types of periodic motions: circular , long - and short-axis elliptical , long - and short-axis radial and inclined radial orbits. Long-axis elliptical orbits and their surrounding tubes have significant features: whilst they keep stars away from the centre, they elongate in the same direction as the density profile. These properties are helpful in the construction of self-consistent equilibria.     

Flattened Jaffe models for galaxies

This paper introduces a class of galactic models which extend Jaffe's spherical models to axisymmetric systems, and then studies the properties of their densities and two-integral even distribution functions. The models have finite total mass and finite densities which, at large distances, decay radially like r ⁻⁴ except on the major axis, and like r ⁻³ on the major axis. The more flattened the galaxy, the stronger is the dependence of the even distribution functions on the angular momenta of its stars. Their distribution functions can be obtained by using the maximum entropy principle or assuming the anisotropy of the models. In particular, some formulae analogous to those of Hunter & Qian are obtained to calculate two-integral odd distribution functions, and they can be applied to obtain the distribution functions under the assumption of anisotropy for the oblate models.     

Integrable models of galactic discs with double nuclei

We introduce a new class of 2D mass models, whose potentials are of Stäckel form in elliptic coordinates. Our model galaxies have two separate strong cusps that form double nuclei. The potential and surface density distributions are locally axisymmetric near the nuclei and become highly non-axisymmetric outside the nucleus. The surface density diverges toward the cuspy nuclei with the law     Our model is sustained by four general types of regular orbits: butterfly , nucleophilic banana , horseshoe and aligned loop orbits. Horseshoes and nucleophilic bananas support the existence of cuspy regions. Butterflies and aligned loops control the non-axisymmetric shape of outer regions. Without any need for central black holes, our distributed mass models resemble the nuclei of M31 and NGC 4486B. It is also shown that the self-gravity of the stellar disc can prevent the double nucleus to collapse.     

Prolate Jaffe models for galaxies

We introduce a class of prolate Jaffe models for elliptical galaxies, which are a further extension of Jaffe's spherical models of axisymmetric elliptical systems, and study the properties of their densities, circular velocities, velocity dispersions and two-integral even distribution functions. The form of the potential allows the density to be expressed simply as a function of the potential and radial coordinate R . The models have finite total mass and their densities at large distances decay radially as r ⁻⁴, except on the major axis, where the densities decay as r ⁻³. It is known from Hunter's formulae that the velocity dispersions for prolate models can be expressed in terms of elementary functions of R and z , unlike those for the oblate Jaffe models recently given by Jiang, and that the prolate models have anisotropic velocity distributions. Thus the prolate models are easier to study than the oblate models. It is also found that the two-integral even distribution functions on the physical boundary of the galaxies increase monotonically with the relative energy, for the prolate models. Furthermore, numerical calculation shows that the two-integral even distribution functions generated from their densities are non-negative, even for very 'squeezed' prolate Jaffe models. However, the edge-on projected surface densities for these prolate models cannot be expressed as simply as for the oblate models.     

Central cusp caused by a supermassive black hole in axisymmetric models of elliptical galaxies

We use numerical simulations to investigate the cusp at the centre of elliptical galaxies caused by the slow growth of a supermassive black hole. We study this problem for axisymmetric models of galaxies with or without rotation. The numerical simulations are based on the 'perturbation particles' method, and use GRAPEs to compute the force arising from the cusp. We study the way in which the density cusp is affected by the initial flattening of the model, as well as the role played by initial rotation. The logarithmic slope of the density cusp is found to be very much insensitive to flattening; as a consequence, we deduce that tangential velocity anisotropy – which supports the flattening – is also of little influence on the final cusp. We investigate, via two different kinds of rotating models, the efficiency with which a rotation velocity component builds within the cusp. A cusp in rotation develops only for models where a net rotation component is initially present at high energy levels. The eventual observation of a central rotational velocity peak in elliptical galaxies has, therefore, some implications for the galaxy dynamical history.