[1] **viXra:1011.0039 [pdf]**
*submitted on 17 Nov 2010*

**Authors:** Guy Moore, Richard Moore

**Comments:** 40 pages

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.

**Category:** Mathematical Physics