Faint shells have been found around a large number of ellipticals (Malin and Carter 1980, 1983; Schweizer, 1980). Quinn (1984) has suggested that shells form around ellipticals when a small disk galaxy is tidally disrupted in a near radial orbit. According to this mechanism the remnants of the nearly two-dimensional disk maintain their dimensionality and form sheets. These sheets will phasewrap around the elliptical, and the shells are seen when the line of sight is tangent to one of these surfaces (Quinn 1984, Hernquist and Quinn 1986a,b,c).

We study the change in shape of the orbit of a satellite sinking (because of dynamical friction) towards the center of a larger galaxy. the galaxy is assumed spherically symmetric with distribution function either isotropic or predominantly radial. the satellite is a softened point mass.

Using Melnikov's method to study the appearance of stochastic orbits in perturbed Stäckel potentials, a correlation is found between the observed shapes of elliptical galaxies and the occurence of mainly regular orbits. Some other potential perturbations giving rise to large regions of stochastic orbits, on the other hand, appear to be inconsistent with observations.

We examine the general instability at large amplitude of the radial periodic orbits along the rotation axis of bulges, spheroids and other rotating triaxial ellipsoidal systems.

We measure quantitatively the growth of stochasticity due to various perturbations to a model of elliptical galaxy. This is achieved by computing the Liapunov characteristic exponents of randomly selected orbits.

Since galaxies are collisionless relaxed systems, actions are an extremely useful tool for understanding their dynamics. There are many potential applications of actions: (1) When orbits in an N-body simulation are characterized by their actions, the six dimensional distribution function, can be reduced to a more tractable three dimensional function, f(J). (2) Actions are adiabatic invariants, and thus are useful for studying slowly evolving systems. Binney, May and Ostriker (1986) have applied this technique to study the response of the spheroid to the disc. (3) the spectral decomposition of an orbit can be used to help generate self–consistent galaxy models (Spergel 1987).

The structure of a galaxy model is described completely by its phase–space distribution function f. By Jeans' Theorem f can be written as a function of the integrals of motion admitted by the potential of the model. Various independent combinations of the integrals may be used as arguments of f; in many cases the action integrals are to be preferred. For a general N–body model, these can be obtained by numerical integration and subsequent spectral decomposition of each orbit (Binney and Spergel 1984).

We have used Schwarzschild's (1979) method to study the variety of self-consistent solutions available to the family of triaxial “perfect ellipsoids” (de Zeeuw 1985). the time-averaged density at 240 points within the mass model is computed for each of 1065 orbits distributed regularly in phase space and covering all four major orbit families. the underdetermined linear system thereby defined is solved in two ways. First, Lucy's (1974) iterative scheme is used to find a “smooth” solution, lying in the interior of the (mathematically-allowed) solution space. Second, linear programming is used, with linear combinations of the x and z components of the total angular momentum as cost functions, to delineate the boundary of the projection of the solution space in the Lx — Lz plane. the above procedure is applied to 21 figures, with axis ratios, b/a and c/a, chosen at equal intervals of 1/8.

A method for the construction of models of axisymmetric galaxies is presented. in this formulation we determine the distribution function corresponding to a given gravitational potential and the associated mass density distribution. Although the realization of the model is numerical, the underlying theory is analytic and exact. This method allows us to construct a wide range of models without having to use linear programming and a large amount of computer time. Here we present the results from the application of this method to the “perfect” oblate-spheroid mass model. A large class of valid self-consistent distribution functions which depend on three isolating integrals of the motion is found. the kinematics of many models are consistent with those observed for elliptical galaxies. in particular, models generated by this formulation are in agreement with the observed values of the ratio of the maximum projected rotational velocity to the velocity dispersion along the line of sight versus ellipticity.

Self-consistent solutions for the perfect elliptic disk have been obtained. Velocity field, dispersion maps and the distribution function in action space have been derived.

One dimensional “needles” are a limiting case of general triaxial stellar systems. Self-consistent, finite needles can have arbitrary longitudinal density distributions but have a fixed, universal distribution function. All needles are stable to all longitudinal perturbations but neutral to transverse perturbations.

Triaxial separable potentials V(λ, μ,ν) = fλ + fμ + fν ≥ 0, with admit three constants of the motion (see de Zeeuw 1985 for more details), for which we can take the following set:

Here vλ, and vν are the velocity components in the (λ, μ,ν) coordinate system, and we recognize in E the binding energy per unit mass.

We consider a family of inhomogeneous axisymmetric mass models with a simple gravitational potential originally introduced by Kuzmin (1956): where R2 = x2 + y2, a and c are parameters, G is the gravitational constant, and M is the total mass of the corresponding density distribution.

Two surveys of dynamical models of highly flattened, triaxial elliptical galaxies with isothermal potentials have been constructed. These models were constructed in order to better understand the range of possible observable dynamical properties of triaxial galaxies. All models have been constructed so that they appear as E6 galaxies when seen from their intermediate axes. However, one set of models is nearly oblate; the other is nearly prolate. the models are constructed with massive halos such that M/L ∝ r. Triaxial models of either shape can be constructed with their projected axes of rotation at any position angle with respect to the major axes of the galaxies. the most surprising result is that in most models, the position angle of maximum observed rotation is not perpendicular to the position angle of zero rotation.

We describe here a way of constructing models for box–shaped dynamical systems without cylindrical rotation. We argue that such models may arise as a result of the external heating of a disk.

The R1/4 luminosity law has often been used as an observational constraint in the construction of self-consistent models of ellipticals. In contrast, in this and in the following paper, developing some ideas proposed in the past (BS8A, SB85), we investigate certain theoretical arguments that may lead to a dynamical justification of this important empirical law.

Elliptical galaxies are expected to have undergone incomplete violent relaxation. Here incomplete relaxation is regarded as a process producing a metastable, long-lived state which is dynamically stabilized by the approximate conservation of one global quantity in addition to the total energy and number of particles.

Distribution functions containing cutoff in energy impose several limitations on systems they describe, e.g., no circular orbits are allowed in the major part of system and spatial boundary is poorly defined. As opposed to these functions, we present here distribution function that describes galaxies of finite extent (i.e., truncated in radius). We discuss properties of systems having this distribution function for spherical and axisymmetric cases and compare their surface brightness, isophotes, and rotation curves with observations of elliptical galaxies. This distribution function can easily be generalized to a triaxial galaxy.

Using a theoretical model for the functional distribution of stars in phase space in a spherically symmetric galactic system, it is found upon solving the fundamental equations of stellar dynamics that the rotation curves produced by the model are flat for large distances within the system. the properties of the stellar orbits within such systems are investigated and an N-ring axially and equatorially symmetric model for simulating its dynamics is presented. Poisson's equation is solved by expanding density and potential in Legendre polynomials (c.f. van Albada and van Gorkom, 1977). It is helped to follow the time development of such a system under various forces.

The light distribution in the envelopes of spherical galaxies seems to be caused by the existence of a break in the energy distribution, N(E) at E = 0. This, in turn is probably caused by the escape of positive energy stars.