**Authors:** Guy Moore, Richard Moore

Our obtaining the analytical equations for the gravitation of a particular type of mathematical annulus, which we called a 'Special Gravitating Annulus' (SGA), greatly facilitates studying its orbital properties by computer programming. This includes isomorphism, periodic and chaotic polar orbits, and orbits in three dimensions. We provide further insights into the gravitational properties of this annulus and describe our computer algorithms and programs. We study a number of periodic orbits, giving them names to aid identification. 'Ellipses extraordinaires' which are bisected by the annulus, have no gravitating matter at either focus and represent a fundamental departure from the normal association of elliptical orbits with Keplerian motion. We describe how we came across this type of orbit and the analysis we performed. We present the simultaneous differential equations of motion of 'ellipses extraordinaires' and other orbits as a mathematical challenge. The 'St.Louis Gateway Arch' orbit contains two 'instantaneous static points' (ISP). Polar elliptical orbits can wander considerably without tending to form other kinds of orbit. If this type of orbit is favoured then this gives a similarity to spiral galaxies containing polar orbiting material. Annular oscillatory orbits and rotating polar elliptical orbits are computed in isometric projection. A 'daisy' orbit is computed in stereo-isometric projection. The singularity at the centre of the SGA is discussed in relation to mechanics and computing, and it appears mathematically different from a black hole. In the Appendix, we prove by a mathematical method that a thin plane self-gravitating Newtonian annulus, free from external influence, exhibiting radial gravitation that varies inversely with the radius in the annular plane, must have an area mass density which also varies inversely with the radius and this exact solution is the only exact solution.

**Comments:** 40 pages

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[v1] 17 Nov 2010

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