A Search for Collision Orbits in the Free-Fall Three-Body Problem II

A numerical procedure to systematically find collision orbits in the planar three-body problem has been developed in the preceding paper (Tanikawa et al., 1995). Using this procedure, a search for binary and triple collision orbits has been carried out in the free-fall three-body problem. Some detailed structures of a part of the initial value space are discussed. Various interesting orbits have been found. Examples are oscillatory orbits in which ejected particles change from ejection to ejection, and orbits which are not isosceles initially but nearly isosceles after escape. Some results of isosceles problems (Simó and Martínez, 1988) are extended to non-isosceles problems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Triple collisions in the one-dimensional three-body problem

We consider the motions of particles in the one-dimensional Newtonian three-body problem as a function of initial values. Using a mapping of orbits to symbol sequences we locate the initial values leading to triple collisions. These turn out to form curves which give clear structure to the region in which the motions depend sensitively on initial conditions. In addition to finding the triple collision orbits we also locate orbits which end up to a triple collision in both directions of time, that is, orbits which are finite both in space and time. This revised version was published online in July 2006 with corrections to the Cover Date.     

Algorithmic regularization of the few-body problem

Using a modified leapfrog method as a basic mapping, we produce a new numerical integrator for the stellar dynamical few-body problem. We do not use coordinate transformation and the differential equations are not regularized, but the leapfrog algorithm gives regular results even for collision orbits. For this reason, application of extrapolation methods gives high precision. We compare the new integrator with several others and find it promising. Especially interesting is its efficiency for some potentials that differ from the Newtonian one at small distances.     

Asymmetric X-ray diffraction peak of metamorphosed carbonaceous material

Abstract It was revealed that the asymmetric X-ray diffraction peak of metamorphosed carbonaceous material was a combination of two types of symmetric peaks from the coaly material and graphite crystallite. Numerical synthesis using the two symmetric peaks shows the graphitization process observed in a metamorphic sequence is accompanied by a continuous change of the peak shape and apparent interplanar spacing. Based on the analytical results and geological observations, it is suggested that graphitization of carbonaceous material consists of two processes: (i) the formation of graphite crystallites at the expense of the coaly material through the formation of transitional material and the increasing abundance of crystallites which is observed in the chlorite zone; and (ii) growth of crystallites forming well-ordered graphite, which occurs in the higher-grade zone.     

Effect of winter meteorological conditions on the formation of the cold bottom water in the eastern Bering Sea shelf

The cold bottom water, formed in the previous winter on the eastern Bering Sea shelf, remains throughout the summer. in order to examine the mechanism for the formation of the cold bottom water, we used minimum water temperature in the cold bottom water observed over the eastern Bering Sea shelf for 30 years. The interannual variation in the minimum water temperature of the cold bottom water was closely related to that of mean air temperature during cooling period at St. Paul Island. The air temperature in previous winter primarily affects the cold bottom water. We estimated decrement of the water temperature due to ice melting with simple box model. It was found with the box model that decreasing of the water temperature and lowering of the salinity depend on ice melting. To investigate the cause of interannual variation in air temperature in winter, we applied EOF analysis to the 500 hPa height. The Pacific/North American pattern (PNA) was related to mean air temperature at St. Paul Island in cooling season and the cold bottom water temperature. These results suggest the connection between ENSO events and warming or cooling in the Bering Sea shelf in winter.     

Orbital Distribution Arbitrarily Close to the Homothetic Equilateral Triple Collision in the Free‐Fall Three‐Body Problem with Equal Masses

The existence of escape and nonescape orbits arbitrarily close to the homothetic equilateral triplecollision orbit is considered analytically in the threebody problem with zero initial velocities and equal masses. It is proved that escape orbits in the initial condition space are distributed around three kinds of isosceles orbits. It is also proved that nonescape orbits are distributed in between the escape orbits where different particles escape. In order to show this, it is proved that the homotheticequilateral orbit is isolated from other triplecollision orbits as far as the collision at the first triple encounter is concerned. Moreover, the escape criterion is formulated in the planarisosceles problem and translated into the words of regularizing variables. The result obtained by us explains the orbital structure numerically.