Use of the Geometric Elements in Numerical Simulations

We derive the transformations to convert the state vector in cartesian coordinates into geometric orbital elements (and conversely the geometric elements into the state vector) for a test particle moving around an oblate planet. These transformations arise from the epicyclic theory and are accurate to second order in eccentricity and inclination. This paper is written to be directly used for computational purposes, such as the numerical study of ring dynamics.     

Merger Dynamics of the Pair of Galaxy Clusters——A399 and A401

Convincing evidence for a past interaction between the two rich clusters A399 and A401 was recently found in the X-ray imaging observations. We examine the structure and dynamics of this pair of galaxy clusters. A mixture-modeling algorithm was applied to obtain a robust partition into two clusters, which allowed us to discuss the virial mass and velocity distribution of each cluster. Assuming that these two clusters follow a linear orbit and they have once experienced a close encounter, we model the binary cluster as a two-body system. As a result, four gravitationally bound solutions are obtained. The recent X-ray observations seem to favor a scenario in which the two clusters with a true separation of 5.4h-1 Mpc are currently expanding at 583 km s-1 along a direction with a projection angle of 67.5°, and they will reach a maximum extent of 5.65 h-1 Mpc in about 1.0 h-1 Gyr.     

Invariant manifolds, phase correlations of chaotic orbits and the spiral structure of galaxies

In the presence of a strong   m = 2  component in a rotating galaxy, the phase-space structure near corotation is shaped to a large extent by the invariant manifolds of the short-period family of unstable periodic orbits terminating at L ₁ or L ₂. The main effect of these manifolds is to create robust phase correlations among a number of chaotic orbits large enough to support a spiral density wave outside corotation. The phenomenon is described theoretically by soliton-like solutions of a Sine–Gordon equation. Numerical examples are given in an N -body simulation of a barred spiral galaxy. In these examples, we demonstrate how the projection of unstable manifolds in configuration space reproduces essentially the entire observed bar–spiral pattern.