Ring Planets in astronomy
Arithmetic has the integral part “rings” in its name. Rings are geometric structures whose elements can be added, subtracted, multiplied and divided. In mathematics, rings are geometrical structures which generalize different fields: addition, subtraction, division need not be commutitive and multiplicative polygons need not exist in a single ring. Thus, a ring is also a group equipped with two concomitant operations satisfying symmetries analogous to those of division and addition of numbers.
The most well-known example of a ring in mathematics is the “ring theory” of prime numbers. Here, every number can be regarded as a prime number at a given point, thus generating a connotative model in a ring format. The ring particles create an orbital around a point, and the central point acts as the base or origin of the ring structure, while the points on the ring surface generate the orbital velocity. The orbit particles move through the interior of the ring, and their orbital motion gives rise to the elements of the ring structure.
The study of rings and their numerical representations goes back to Dedekind. His complete model of the arithmetic interval over a finite range was realized using a mathematical ring map. The ring map was first applied to the elliptical orbit formulas by Grover. More recent work by Holt shows how the ring map could be used to cohominate the Geometry of concentric geometrical spaces. The rings can be studied on the plane of tangent lines and on the hyperbolic plane. The semi-spherical model of a ring can be studied on the half-plane of tangent lines and the hyperbolic plane.
A similar approach is applied to the study of elliptical orbit elements, with Studer and van der Grint showing that the radii of the inner and outer orbits are the same at every angle. They derive the formula for the inner radii of an elliptical orbit by taking the mean values of the components of the tangent plane’s algebraic equations. Studer also derives the formulas for inner and outer radii of semi-spherical and hyperbolic planes by means of integral functions. van der Grint found the same functions for the radii of elliptical rings.
Orbit construction is also based on the formulas for the radii and periods of the inner and outer orbits when they are spherical, semi-spherical, and hyperbolic. It is found that there are a number of equators whose periods are the same throughout the disk. The period of one equator is proportional to the radius of the disk, whilst the period of the other equator is equal to the eccentricity of the orbit. Similarly the inner and outer semi-spheres have definite periods that depend solely upon the eccentricity. Studer and van der Grint give the formula for calculating these two periods and their derivatives.
The Voyage to the Center of the Earth also includes calculations for the inner and outer radial axes and the periods of the major equatorial rings. It is found that the equator has only one ring with a semi-sphere whose period is the same as the mean solar orb. The inner circle of disks has five spokes while the outer ring has only three spokes. This can be compared to the Berenstein Bears which have six spokes.