N-body simulations using customized potential–density pair basis sets

Potential–density pair basis sets can be used for highly efficient N -body simulation codes, but they suffer from a lack of versatility, i.e. a basis set has to be constructed for each different class of stellar system. We present numerical techniques for generating a biorthonormal potential–density pair basis set that has a general specified pair as its lowest-order member. We go on to demonstrate how the set can be used to construct N -body equilibria, which we then evolve using an N -body code that calculates forces using the basis set.     

Stability of power-law discs — I. The Fredholm integral equation

The power-law discs are a family of infinitesimally thin, axisymmetric stellar discs of infinite extent. The rotation curve can be rising, falling or flat. The self-consistent power-law discs are scale-free, so that all physical quantities vary as a power of radius. They possess simple equilibrium distribution functions depending on the two classical integrals, energy and angular momentum. While maintaining the scale-free equilibrium force law, the power-law discs can be transformed into cut-out discs by preventing stars close to the origin (and sometimes also at large radii) from participating in any disturbance. This paper derives the homogeneous Fredholm integral equation for the in-plane normal modes in the self-consistent and the cut-out power-law discs. This is done by linearizing the collisionless Boltzmann equation to find the response density corresponding to any imposed density and potential. The normal modes — that is, the self-consistent modes of oscillation — are found by requiring the imposed density to equal the response density. In practice, this scheme is implemented in Fourier space, by decomposing both imposed and response densities in logarithmic spirals. The Fredholm integral equation then relates the transform of the imposed density to the transform of the response density. Numerical strategies to solve the integral equation and to isolate the growth rates and the pattern speeds of the normal modes are discussed.     

Analytical potential–density pairs from complex-shifted Kuzmin–Toomre discs

The complex-shift method is applied to the Kuzmin–Toomre family of discs to generate a family of non-axisymmetric flat distributions of matter. These are then superposed to construct non-axisymmetric flat rings. We also consider triaxial potential–density pairs obtained from these non-axisymmetric flat systems by means of suitable transformations. The use of the imaginary part of complex-shifted potential–density pairs is also discussed.     

Exact density–potential pairs from complex-shifted axisymmetric systems

In a previous paper, the complex-shift method has been applied to self-gravitating spherical systems, producing new analytical axisymmetric density–potential pairs. We now extend the treatment to the Miyamoto–Nagai disc and the Binney logarithmic halo, and we study the resulting axisymmetric and triaxial analytical density–potential pairs; we also show how to obtain the surface density of shifted systems from the complex shift of the surface density of the parent model. In particular, the systems obtained from Miyamoto–Nagai discs can be used to describe disc galaxies with a peanut-shaped bulge or with a central triaxial bar, depending on the direction of the shift vector. By using a constructive method that can be applied to generic axisymmetric systems, we finally show that the Miyamoto–Nagai and the Satoh discs, and the Binney logarithmic halo cannot be obtained from the complex shift of any spherical parent distribution. As a by-product of this study, we also found two new generating functions in closed form for even and odd Legendre polynomials, respectively.     

Exact density–potential pairs from the holomorphic Coulomb field

We show how the complex-shift method developed by Appell to study the gravitational field of a point mass (and used in electrodynamics by, among others, Newman, Carter, Lynden-Bell, and Kaiser to determine some remarkable properties of the electromagnetic field of rotating charged configurations) can be extended to obtain new and explicit density–potential pairs for self-gravitating systems departing significantly from spherical symmetry. The rotational properties of two axisymmetric baroclinic gaseous configurations derived with the proposed method are illustrated.     

On the potential–density relation for Stäckel discs

The general form of the surface density of an infinitely thin disc is given that generates a Sta¨ckel potential in the disc only, using formulae for the potential of elliptic and hyperbolic strings. This is useful for problems in which a simple form for the potential is important, while the corresponding surface density need only be known to check (numerically) that it is positive. A simple potential with a positive surface density is given. Also, formulae are given to calculate the surface density of such a Sta¨ckel disc, in the case in which the rotation curve is given and all the mass is concentrated in the disc.     

Simulations of spheroidal systems with substructure: trees in fields

We present a hybrid technique of N -body simulation to deal with collisionless stellar systems having an inhomogeneous global structure. We combine a treecode and a self-consistent field code such that each of the codes models a different component of the system being investigated. The treecode is suited to treatment of dynamically cold or clumpy components, which may undergo significant evolution within a dynamically hot system. The hot system is appropriately evolved by the self-consistent field code. This combined code is particularly suited to a number of problems in galactic dynamics. Applications of the code to these problems are briefly discussed.     

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.     

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.     

Invariant distributions and collisionless equilibria

This paper discusses the possibility of constructing time-independent solutions to the collisionless Boltzmann equation which depend on quantities other than global isolating integrals such as energy and angular momentum. The key point is that, at least in principle, a self-consistent equilibrium can be constructed from any set of time-independent phase-space building blocks which, when combined, generate the mass distribution associated with an assumed time-independent potential. This approach provides a way to justify Schwarzschild's method for the numerical construction of self-consistent equilibria with arbitrary time-independent potentials, generalizing thereby an approach developed by Vandervoort for integrable potentials. As a simple illustration, Schwarzschild's method is reformulated to allow for a straightforward computation of equilibria which depend only on one or two global integrals and no other quantities, as is reasonable, for example, for modelling axisymmetric configurations characterized by a non-integrable potential.     

Analytical potential–density pairs for flat rings and toroidal structures

The Kuzmin–Toomre family of discs is used to construct potential–density pairs that represent flat ring structures in terms of elementary functions. Systems composed of two concentric flat rings, a central disc surrounded by one ring and a ring with a centre of attraction are also presented. The circular velocity of test particles and the epicyclic frequency of small oscillations about circular orbits are calculated for these structures. A few examples of three-dimensional potential–density pairs of 'inflated' flat rings (toroidal mass distributions) are presented.     

Algorithmic regularization of the few-body problem

Using a modified leapfrog method as a basic mapping, we produce a new numerical integrator for the stellar dynamical few-body problem. We do not use coordinate transformation and the differential equations are not regularized, but the leapfrog algorithm gives regular results even for collision orbits. For this reason, application of extrapolation methods gives high precision. We compare the new integrator with several others and find it promising. Especially interesting is its efficiency for some potentials that differ from the Newtonian one at small distances.     

Orbit classification in arbitrary 2D and 3D potentials

A method of classifying generic orbits in arbitrary 2D and 3D potentials is presented. It is based on the concept of spectral dynamics introduced by Binney &38; Spergel that uses the Fourier transform of the time series of each coordinate. The method is tested using a number of potentials previously studied in the literature and is shown to distinguish correctly between regular and irregular orbits, to identify the various families of regular orbits (boxes, loops, tubes, boxlets, etc.), and to recognize the second-rank resonances that bifurcate from them. The method returns the position of the potential centre and, for 2D potentials, the orientation of the principal axes as well, should this be unknown. A further advantage of the method is that it has been encoded in a FORTRAN program that does not require user intervention, except for 'fine tuning' of search parameters that define the numerical limits of the code. The automatic character makes the program suitable for classifying large numbers of orbits.     

The Gyldén‐type problem revisited: More refined analytical solutions

We resume and consistently extend our previous researches concerning the Gyldén‐type problem (a two‐body problem with time‐dependent equivalent gravitational parameter). To approach most of the concrete astronomical situations to be modelled in this way, we consider a periodic small perturbation. For the nonresonant case, we present a second‐order analytical solution. For the resonant case, we adopt the most realistic astronomical situation: only one dominant term of the Hamiltonian. In this case we point out a fundamental model of resonance, common to every resonant situation, and, moreover, identical to the first fundamental model of resonance . Considering the simplest model of periodic change of the equivalent gravitational parameter, we .nd that all possible resonances are con.ned to the first fundamental model. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Gauss's method for secular dynamics, softened

We show that the algorithm proposed by Gauss to compute the secular evolution of gravitationally interacting Keplerian rings extends naturally to softened gravitational interactions. The resulting tool is ideal for the study of the secular dynamical evolution of nearly Keplerian systems such as stellar clusters surrounding black holes in galactic nuclei, cometary clouds or planetesimal discs. We illustrate its accuracy, efficiency and versatility on a variety of configurations. In particular, we examine a secularly unstable system of counterrotating discs, and follow the unfolding and saturation of the instability into a global, uniformly precessing, lopsided  ( m = 1)  mode.