3-Body Problems, Hidden Constants, Trojans and WIMPs

This work includes two new results - principally two new constants of motion for the linearised restricted 3-body problem (e.g. for the Trojan asteroids) and an important isosceles triangle generalisation of Lagrange's equilateral triangle solution of the restricted case leading to hidden constants for Hildans as well as Trojans. Both of these results are classical, but we also have included new results on Newtonian quantum gravity emanating from the asymptotics relevant for WIMPish particles, explaining the origin of systems like that of the Trojans. The latter result uses a generalisation of our semi-classical mechanics for Schr\"odinger equations involving vector as well as scalar potentials, presented here for the first time, thereby providing an acid test of our ideas in predicting the quantum curvature and torsion of WIMPish trajectories for our astronomical elliptic states. The combined effect is to give a new celestial mechanics for WIMPs in gravitational systems as well as new results for classical problems. As we shall explain, we believe these results could help to see how spiral galaxies evolve into elliptical ones. A simple classical consequence of our isosceles triangle result gives a Keplerian type $4^{\textrm{th}}$ Law for 3-body problems. This is confined to the Appendix.Comment: 43 pages, no figure

The Divine Clockwork: Bohr's correspondence principle and Nelson's stochastic mechanics for the atomic elliptic state

We consider the Bohr correspondence limit of the Schrodinger wave function for an atomic elliptic state. We analyse this limit in the context of Nelson's stochastic mechanics, exposing an underlying deterministic dynamical system in which trajectories converge to Keplerian motion on an ellipse. This solves the long standing problem of obtaining Kepler's laws of planetary motion in a quantum mechanical setting. In this quantum mechanical setting, local mild instabilities occur in the Kelperian orbit for eccentricities greater than 1/\sqrt{2} which do not occur classically.Comment: 42 pages, 18 figures, with typos corrected, updated abstract and updated section 6.

The effects of moisture on mountain lee waves

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Meteorology and Physical Oceanography, 1981.MICROFICHE COPY AVAILABLE IN ARCHIVES AND LINDGREN.Bibliography: leaves 136-139.by Dale Richard Durran.Ph.D

The triggering of orographic rainbands by small-scale topography

The triggering of convective orographic rainbands by small-scale topographic features is investigated through observations of a banded precipitation event over the Oregon Coastal Range and simulations using a cloud-resolving numerical model. A quasi-idealized simulation of the observed event reproduces the bands in the radar observations, indicating the model’s ability to capture the physics of the band-formation process. Additional idealized simulations reinforce that the bands are triggered by lee waves past small-scale topographic obstacles just upstream of the nominal leading edge of the orographic cloud. Whether a topographic obstacle in this region is able to trigger a strong rainband depends on the phase of its lee wave at cloud entry. Convective growth only occurs downstream of obstacles that give rise to lee-wave-induced displacements that create positive vertical velocity anomalies w_c and nearly zero buoyancy anomalies b_c as air parcels undergo saturation. This relationship is quantified through a simple analytic condition involving w_c, b_c, and the static stability N_m^2 of the cloud mass. Once convection is triggered, horizontal buoyancy gradients in the cross-flow direction generate circulations that align the bands parallel to the flow direction