22-3 Aubrey Truman, Richard Durran, Andrew Neate
On Newtonian Quantum Gravity for Stationary States of WIMPs and Celestial Mechanics (1695K, PDF) Jan 10, 22
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Abstract. We discuss the leading term in the semi-classical asymptotics of Newtonian quantum gravity for the Kepler problem. For dark matter, ice or dust particles in the gravitational eld of a star or massive planet this explains how rapidly planets or ring systems can be formed by considering semi-classical equations for the asymptotics of the relevant Schrodinger wave function. We extend this treatment to isotropic harmonic oscillator potentials in two and three dimensions nding explicit solutions, with important applications to the Trojan asteroids. Further in two dimensions, we nd the explicit solutions for Newton's corresponding revolving orbits, explaining planetary perihelion advance in these terms. A general implicit approach to solving our equations is given using Cauchy characteristic curves giving necessary and sufficient conditions for existence and uniqueness of our solutions. Using this method we solve the two dimensional Power Law Problem in our setting. Our methods give an insight to the behaviour of semi-classical orbits in the neighbourhood of classical orbits showing how complex constants of the motion emerge from the SO(4) symmetry group and what the particle densities have to be for circular spirals for different central potentials. Such constants have an important role in revolving orbits. Moreover, they and the aforementioned densities could give rise to observable effects in the early history of planetary ring systems. We believe the quantum spirals here associated with classical Keplerian elliptical orbits explain the evolution of galaxies, with a neutron star or black hole at their centre, from spiral to elliptical.

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