We analyze a class of spherical stellar systems with an isotropic, non-isothermal, one-particle distribution function which is truncated in energy ε as (εb —ε)ν (ν > 0) near the energy of truncatingεb and is the Maxwell-Boltzmannian at |ε| ≫ |εb|. the stellar systems which correspond to this distribution function appear to be isothermal in their central parts and polytropic in the external parts. Some qualitative results which follow from such a “polytropic-isothermal” distribution function (PIDF) are briefly discussed. A further extention of PIDF enables us to analyze in a new way the evolution of spherical stellar systems.

We consider the linear stability of spherical stellar systems by solving the Vlasov and Poisson equations which yield a matrix eigenvalue problem to determine the growth rate. We consider this for purely growing modes in the limit of vanishing growth rate. We show that a large class of anisotropic models are unstable and derive growth rates for the particular example of generalized polytropic models. We present a simple method for testing the stability of general anisotropic models. Our anlysis shows that instability occurs even when the degree of anisotropy is very slight.

We use N-body simulations to study the time evolution and the final shape of the density profiles of non-rotating spherical galaxies that have undergone a tidal encounter. We consider models with de Vaucouleurs and King surface density profiles and with isotropic, tangential and radially biased velocity distributions.

We investigate whether dissipationless collapse starting from very cold, non-rotating initial conditions can produce objects resembling real elliptical galaxies. We also study the effect of various initial geometries on the shape of the final object. Collapses that are initially very cold (2T/W < 0.1) are different from warmer collapses, due to the presence of a dynamical instability associated with clumping of nearly-radial orbits (Polyachenko 1981). This instability can produce very elongated bars (1.6 to 2.1 axis ratio) from spherical initial conditions. the instability is also present in models evolved from oblate and triaxial initial conditions. Warm collapses tend to preserve their initial shapes. Cold initial conditions produce objects whose surface density profiles are well fit by a de-Vaucouleurs law; warm collapses, on the other hand, produce a core-halo profile. A large collapse factor seems necessary to produce objects resembling real galaxies; the same collapse factor guarantees the presence of the radial orbit instability. It thus appears that initial flattening is not crucial for producing prolate or nearly prolate galaxies. Oblate galaxies, on the contrary, seem very difficult to form, unless extremely flattened initial conditions are invoked. Preliminary experiments suggest that these results are not changed by realistic amounts of angular momentum.

We have used the available published observations of the rotational properties of elliptical galaxies to test theories of galaxy formation which predict an anti-correlation between the angular momentum of a galaxy and its initial overdensity and hence formation epoch. We find that the prediction, at least in its simplest form, is not supported by the data for well studied elliptical galaxies which have a range in rotational support.

The one dimensional gravitational model consists of N mass sheets with surface density mi, parallel to the (y, z)–plane and constrained to move along the x-axis under influence of their mutual gravitational force Fij = −2πGmimj sgn(xi – xj). in order to study the evolution of this one–dimensional system, the N Newtonian equations of motion are integrated numerically, using an “exact” double precision algorithm.

Using the second order tensor virial equations and the equation for the rate of change of the kinetic energy tensor, we follow the dynamical evolution of a star cluster with anisotropic velocity distribution. We show that the cluster executes finite amplitude oscillations both in size and eccentricity. However, unlike in the isotropic case, the amplitude now depends on the initial eccentricity.

We use a three-component model of a spiral galaxy, given by (Huang et al., 1979):

1. 1. An outer halo with a gravitational potential of the form where R2 = x2 + y2 + z2, M1 = 1.2 × 1011M⊙ and b1 = 1.1 Kpc.

2. 2. A nucleus, with a gravitational potential given by where M0 = 1.1 × 107M⊙, and b0 = 0.61 × 10−2 Kpc.

3. 3. A self-gravitating disk containing N stars. The density distribution in the z-direction is assumed to be: where α = 2.1 Kpc−1 is the equivalent semi-thickness of the galaxy, and σ(r) is the surface density. We consider three different initial expressions for σ(r):

   1. i. A Toomre (1963) disk.

   2. ii. A uniform distribution.

   3. iii. An exponential distribution.

Applying a criterion previously derived by the author, the stability of stellar systems with an isotropic distribution velocity distribution and Emden's polytropic density law is demonstrated for the exponent n = 3/2.

Models of spherical, self-gravitating stellar systems in which all the trajectories pass initially near the center are interesting to examine from different perspectives. Such models are compatible not only with the idea of explosive cosmogony, but also with the concept of formation of stars from a gaseous, spherical cloud, if they are not affected during formation by significant peculiar motions and thus “fall” into the center. Another question is whether such a system is capable of remaining stable for a long period of time. We will show that such a system is unstable with respect to regular forces and must be reconstructed during the course of one period of revolution of an individual star.

As is well known, in the case of an axially symmetric and time-invariant gravitational potential, if the potential satisfies one particular additional constraint, there exist three isolating integrals of motion: the energy integral, the area integral, and the third integral which is quadratic in the velocities. This work discusses the case in which there exist quadratic integrals in the absence of axial symmetry of the potential. Such a case has already been examined by Eddington [1], but in their explicit form, the integrals were introduced by Clark [2].