Myron Lecar

On the original distribution of the asteriods. I.

Abstract The depletion of an initially uniform distribution of asteroids extending form Mars to Saturn, caused by the gravitational perturbations of Jupiter and Saturn, is calculated by numerical integration of the asteroid orbits. Almost all (about 85%) the asteroids between Jupiter and Saturn are ejected in the first 6000 years Most of the asteroids between the 2 3 Jupiter resonance (4.0 A.U.) and Jupiter are ejected in the first 2400 years with the exception of the stable librators (e.g., the Hilda group). Interior to the 2 3 resonance the depletion was small, and interior to the 1 2 resonance (3.3 A.U.) no asteroids were ejected in the first 2400 years.

A model for close encounters in the planetary problem

Abstract A model is proposed for single close encounters between two small masses, m 1 and m 2 , which orbit a much larger mass, M . The main new feature of the model is the assumption of conic motion of the center of mass of m 1 and m 2 in the gravitational field of M . Comparisons of the model with the three-body equations of motion indicate that the model is a useful approximation for m 1 , m 2 ≲ 10 −5 M . The model is therefore applicable for encounters between bodies of the order of an earth mass or smaller in the presence of the sun. Comparisons are also made of outcomes obtained by the model with outcomes of numerical integration for a large variety of close encounters. The above comparisons reveal that for many purposes the model is an adequate approximation for those encounters with ϵ ≥ 4, where ϵ is the eccentricity of the hyperbolic orbit of m 1 about m 2 .

Numerical Experiments on Lynden-Bell’s Statistics

We performed computer experiments on 13 different initial configurations of one-dimensional self-gravitating systems. The three most and the three least violently relaxed systems were compared with the predictions of Lynden-Bells statistical mechanics. The agreement between the experimental results and the theoretical predictions became worse as the relaxation became more violent. While all six systems were theoretically nondegenerate, the violent systems invariably flung out a halo that took most of the energy, leaving behind a low-energy degenerate core.

The Collective Relaxation of Two-Phase-Space-Density Collisionless One-Dimensional Selfgravitating Systems

The evolution of one-dimensional two-phase-space-density selfgravitating systems of stars is investigated by following the motion of the boundary curves of the systems in phase space.

A Phase-Space Boundary Integration of the Vlasov Equation for Collisionless One-Dimensional Stellar Systems

The evolution of one dimensional (stratified) self-gravitating systems of stars with constant phase-space density (‘water bag’ model) is investigated by following the motion of the boundary curves defining the systems. The results are compared with those obtained by sheet-model computer experiments and good agreement is found. New aspects of the evolution, revealed by the present method, are discussed.

COLLECTIVE RELAXATION OF TWO-PHASE-SPACE-DENSITY COLLISIONLESS ONE- DIMENSIONAL SELFGRAVITATING SYSTEMS.

The evolution of one-dimensionaltwo-phase-space-density selfgravitating systems of stars is investigated by following the motion of the boundary curves of the systems in phase space.Aqualitative agreement with Lynden-Bells theory predicting, for the most probably state, velocity dispersions inversely proportional to the phase space density of the component at the star formation, is found.

A method of computing the gravitational field of an axially symmetric flat galaxy

A method has been developed for computing the gravitational force field of an axially symmetric flat galaxy from its surface mass density. The method is simple to program, fast, and accurate. An inversion formula is derived that allows computation of surface density from rotation curves by use of any method that converts density to force. The method is compared with a method of Clutton-Brock that utilizes Hankel transforms of Laguerre functions.

Galactic mass determinations from incomplete rotation curves

Given an incomplete rotation curve of a spiral galaxy, various assumptions about the Galaxy beyond the last observed point are made: (A) the force falls off as 1/r2, (B) the mass density is zero, and (C) the mass density falls off as 1/r3. The msss distributions obtained from each of these assumptions are all well behaved, and it is impossible to choose the correct curve from considerations of the resulting mass distributions alone. The correct mass distribution in the disk system of a galaxy cannot be deduced from an incomplete rotation curve.

A Numerical Experiment on Relaxation Times in Stellar Dynamics

The deflection of the velocity vector of a massless test star in the field of 100 stars was determined by numerical integration. The deflection due to each field star independently (with the other field stars removed) was also determined. The square of the deflection caused by the combined action of the field stars agreed quantitatively with the sum of the sauares of the individual deflections and also with the theoretical estimate of Williamson and Chandrasekhar.