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The results are presented of an N-body simulation which shows that it is possible for a highly triaxial self-gravitating system in dynamical equilibrium to exhibit stable figure rotation about its intermediate axis. The system is long-lived, lasting at least 20 half-mass crossing times of the final system. The nature of the simulation is summarized.
The problem is studied of determining a stochastic optimal controller for an n-body spacecraft. The approach used in obtaining the stochastic controller involves the application, interpretation, and combination of advanced dynamical principles and the theoretical aspects of modern control theory. The stochastic controller obtained for a complicated model of a spacecraft uses sensor angular measurements associated with the base body to obtain smoothed estimates of the entire state vector, can be easily implemented, and enables system performance to be significantly improved.
Collisions of galaxies are often invoked to explain violent phenomena in the universe. The dynamics of interacting galaxies is intrinsically three-dimensional and involves both gas and stellar dynamics. In general, a computational approach is needed to model galactic collisions. Galaxy encounters are studied using a hybrid N-body/hydrodynamics code, capable of integrating systems of stars, gas, and dark matter in a fully self-consistent manner. These experiments demonstrate that gravitational coupling between gas and stars in galactic interactions can drive most of the gas throughout a galaxy into the nucleus of a merger remnant. The high densities in these gas concentrations are likely to result in strong bursts of star formation. Hence, this process may explain the nuclear starbursts in some systems of interacting galaxies. Further collapse of these gas concentrations can trigger even more intense activity if some gas is eventually accreted by a supermassive black hole. Such an evolutionary sequence may account for some quasars and active galactic nuclei.
In order to address the early stages of nebula evolution, a three-dimensional collapse code which includes not only hydrodynamics and radiative transfer, but also the effects of ionization and, possibly, magnetic fields is being addressed. As part of the examination of solar system evolution, an N-body code was developed which describes the latter stages of planet formation from the accretion of planetesimals. To test the code for accuracy and run-time efficiency, and to develop a stronger theoretical foundation, problems were studied in orbital dynamics. A regional analysis of the correlation in the gravity and topography fields of Venus was performed in order to determine the small and intermediate scale subsurface structure.
The stability of a set of models based on isothermal oblate E6 elliptical galaxies is studied using N-body techniques. The only stable models are those that are near the isotropic model and have nearly equal number of stars in retrograde and prograde orbits. Fast rotators are unstable to modes that appear to be analogous to the classical streaming instability seen in many disk systems. Systems with a large velocity dispersion in the direction of the cylindrical radius are unstable to modes that appear to be similar to the radial orbit instability observed in some spherical systems. However, evidence is presented that these two instabilities may be related, and an instability criterion that applies to both is constructed.
A self-consistent, three-dimensional numerical experiment is performed on an N-body system whose initial state is a realization of a certain theoretical model of a rotating triaxial galaxy. The model is a stellar-dynamical counterpart of a uniformly rotating polytrope of index equal to 0.5. The aim of the experiment is to study the equilibrium of the system and, in particular, to test its stability. The experimental system behaves in the mean like a realization of the theoretical model for at least seven crossing times. The principal departure of the system from equilibrium is an oscillation which is identified as a radial pulsation. There is no indication in its behavior that the system is unstable with respect to anu mode with an e-folding time shorter than or of the order of two crossing times. Certain changes that occur in the state of the system are interpreted, with the aid of the theoretical model, as secular changes which result from a slight failure of our numerical methods to conserve the mass, energy, and angular momentum of the system; these effects are small enough that they do not vitiate the experiment on a dynamical time scale.
Investigations of the collisional evolution of particulate disks subject to the gravitational perturbation of a more massive particle orbiting within the disk are underway. Both numerical N-body simulations using a novel collision algorithm and analytical kinetic theory are being employed to extend our understanding of perturbed disks in planetary rings and during the formation of the solar system. Particular problems proposed for investigation are: (1) The development and testing of general criteria for a small moonlet to clear a gap and produce observable morphological features in planetary rings; (2) The development of detailed models of collisional damping of the wavy edges observed on the Encke division of Saturn's A ring; and (3) The determination of the extent of runaway growth of the few largest planetesimals during the early stages of planetary accretion.
Modes of oscillation in six rotating, axisymmetric N-body systems are studied in a sequence of self-consistent, three-dimensional numerical experiments. The experimental systems are realizations of theoretical models of galaxies which are stellar-dynamical counterparts of uniformly rotating polytropes of index equal to 0.5. The ratio of the rotational kinetic energy to the gravitational potential energy ranges from 0.13 to 0.20. The systems oscillate axisymmetrically; the oscillations are interpreted as superpositions of a mode of radial pulsation and a Kelvin-like mode of oscillation. The experimental frequencies of these modes agree very well with theoretical predictions. When these modes are suppressed, the states of the experimental systems are very steady. The systems are dynamically unstable with respect to a toroidal mode when the ratio of the rotational kinetic energy to the gravitational potential energy exceeds a value lying between 0.16 and 0.17.
The role of orbit crowding and cloud-cloud collisions in the formation of GMCs and their organization in global spiral structure is investigated. Both N-body simulations of the cloud system and a detailed analysis of individual particle orbits are used to develop a conceptual understanding of how individual clouds participate in the collective density response. Detailed comparisons are made between a representative cloud-particle simulation in which the cloud particles collide inelastically with one another and give birth to and subsequently interact with young star associations and stripped down simulations in which the cloud particles are allowed to follow ballistic orbits in the absence of cloud-cloud collisions or any star formation processes. Orbit crowding is then related to the behavior of individual particle trajectories in the galactic potential field. The conceptual picture of how GMCs are formed in the clumpy ISMs of spiral galaxies is formulated, and the results are compared in detail with those published by other authors.
During the formation of stellar systems such as globular clusters, low-mass subcondensations which eventually form stars must retain a geometric size throughout the collapse process that is small compared to the characteristic distance separating them. If the local velocity dispersion of the subcondensations is small, the overall dimension of the system can decrease substantially before reaching a dynamical equilibrium state. The maximum collapse factor is deduced by examining the growth of the velocity dispersion and the spread in arrival times at the origin caused by local and global fluctuations. It is shown, analytically as well as in a series of N-body simulations, that the maximum reduction in the characteristic dimension of a system of N fragments with an initial homogeneous distribution subject to N exp 1/2 fluctuations is proportional to N exp 1/3. Direct physical collisions between low-mass subcondensations are therefore unlikely to occur in protoglobular clusters. The results are discussed in the context of fragmentation and violent relaxation.
A study to demonstrate how the dynamics of galaxies may be investigated through the creation of galaxies within a computer model is presented. The numerical technique for simulating galaxies is shown to be both highly efficient and highly robust. Consideration is given to the anatomy of a galaxy, the gravitational N-body problem, numerical approaches to the N-body problem, use of the Poisson equation, and the symplectic integrator.
Numerical studies were performed on the dynamical effects on the evolution of clumps, filaments, voids, and galaxy clusters by various final Omega values. The final Omega values examined ranged from 0.03-1, and attention was given to defining observations of superclusters which would aid in determining an actual value for Omega. The numerical trials consisted of n-body integration programs governed by the total expansion and final Omega value and included comparisons between results for open and closed universes. All runs started from the same initial conditions. The dispersion trajectories of particles and the final galactic cluster forms were found to be equivalent, regardless of the final Omega value. The possibility of deriving a value for Omega from velocity data on galaxies which have not yet joined clusters is discussed.
A multiple three-body technique is used to study the orbital evolution of satellite galaxies which is similar to the N-body method but neglects two-body forces between stars in the halo of the parent galaxy. It is found that, for satellites orbiting within the halo, Chandrasekhar's (1960) dynamical friction formula accurately describes the orbital decay rate, including its variation with satellite mass and size and with the number density and mass of halo stars. Significant frictional forces are present even outside the halo, and the orbital decay rate, instead of depending on the procedure used to place the satellite in its orbit, is determined only by the current orbital parameters. This semirestricted N-body method is sufficiently fast to have permitted the running of 200 simulations to date, many more than would have been possible by means of the conventional N-body technique.
Numerical models permit modelling processes within galaxies that are seemingly beyond the reach of physical contact or close-up viewing from different angles. N-body equations permit simple computations of the interactions of individual stars within galaxies, but the motions of these objects feature nonlinear characteristics because of the forces exerted by the complex as a whole which are not amenable to examination with Newtonian laws of force. The models have, however, revealed that thin disk galactic are not stable forms, that barlike structures rotating end-over-end are more stable structures, and that galactic contraction occurs during collisions between galaxies. Internal effects have been successfully predicted for galaxies which are part of clusters, growth rate disturbances due to the expansion of the Universe have been quantified as 2-3 percent and dynamical friction has been shown to damage galaxies which are sliding into background material.
Many carefully selected samples of interacting galaxies have been observed extensively in attempts to clarify whether interaction produces activity in galaxies. Because the sample members represent a wide range of encounter parameters and times, one can then study whether there are correlations between observable encounter features and, for example, Seyfert activity. On the other hand, in theoretical studies, simulations typically deal with either time-consuming detailed modelling of single galaxy pairs or tracing a few model encounters over time. The authors extend the observational survey approach by combining it with a simulation survey. The authors are conducting a survey of model encounters, covering the most important encounter parameters over a wide range. Some parameters, such as companion structure and initial velocity, are demonstratably less important and can be ignored in a first pass. The parameter range must be richly enough sampled so that the authors can evaluate the uniqueness of the observable morphology and velocity structure of the resulting simulated pairs to diagnose unobservable companion orbit parameters. They are using a self-gravitating polar n-body code run on the Cray X-MP at the Alabama Supercomputer Network. For each simulation, the authors have stellar and gas distributions predicted over, typically, a billion years, along with information on gas motions within the disk and any material captured by the companion or lost to the system. Features of disturbed spiral galaxies are sensitive enough to time and encounter parameters so that a match of the simulation survey results to observations can be applied as starting points to infer unobservable orbital or system parameters in actual sample members. This should enable them to examine whether interesting observed properties (Seyfert activity, nuclear star-formation rate) are functions of unobservable dynamical properties which characterize each encounter. Any correlations (or lack of some expected ones) will provide strong clues as to how or whether these phenomena are related to interactions. Aside from its use with such observed samples, this survey should greatly speed determination of initial orbital parameters for more detailed subsequent simulations of individual systems.
A brief description is presented of a canonical set of equations which governs the behavior of an n-body spacecraft. General results are given for the case in which the spacecraft is modeled in terms of n rigid bodies connected by dissipative elastic joints. The final equations are free from constraint torques and involve only r variables (r is the number of degrees of freedom of the system). An advantage which accompanies the elimination of the constraint torques is a decrease in the computer run time (especially when n is large).
For a vehicle amenable to idealization as a set of n + 1 rigid bodies interconnected by n line hinges (implying tree topology), a set of minimum-dimension matrix equations of motion is recorded in terms of variables including the relative rotations of contiguous bodies. When a substructure or portion of the vehicle is subjected to large gross deformations but is restricted to 'small' strains or local deformations, the n-body model is (for sufficiently large n) assumed to experience 'small' relative rotations of contiguous bodies within the substructure, and the equations of motion are linearized in these variables. The application of coordinate transformations and truncations to the linearized subset of the system equations of motion then produces a hybrid-coordinate formulation of the problem involving both discrete and distributed coordinates.
A unified treatment of different reaction mechanisms in nonrelativistic N-body scattering is presented. The theory is based on connected kernel integral equations that are expected to become compact for reasonable constraints on the potentials. The operators T/sub +-//sup ab/(A) are approximate transition operators that describe the scattering proceeding through an arbitrary reaction mechanism A. These operators are uniquely determined by a connected kernel equation and satisfy an optical theorem consistent with the choice of reaction mechanism. Connected kernel equations relating T/sub +-//sup ab/(A) to the full T/sub +-//sup ab/ allow correction of the approximate solutions for any ignored process to any order. This theory gives a unified treatment of all few-body reaction mechanisms with the same dynamic simplicity of a model calculation, but can include complicated reaction mechanisms involving overlapping configurations where it is difficult to formulate models.