Perihelion motion

We now examine the secular motions caused by the electric field. The perihelion of the orbital ellipse alters its position relatively to the line of nodes, and the latter itself moves uniformly about the axis of the field. It follows from (5) that two periods of the perihelion motion occur during one revolution of the line of nodes. 

We still have the two limiting cases of the perihelion motion to... 

In Sect. 1.4, we will demonstrate the validity of the method by analysing the relativistic Kepler problem by computing the perihelion motion of the planet Mercury, followed by Sect. 1.5, displaying the explicit connection between the Schwarzschild singularity and Gddel s theorem. The final conclusion summarises the modus operandi and its subsequent consequences. 

A <1) an elliptic type orbit, cf. the classical case. The latter condition corresponds to a rosette orbit comprising an ellipse with a perihelion motion matching maximum... 

In conclusion, we emphasise the following points (i) we have re-derived a previously obtained operator array formulation, which in its complex symmetric form permits a viable map of gravitational interactions within a combined quantum-classical structure (ii) the choice of representation allows the implementation of a global superposition principle valid both in the classical as well as the quantum domain (iii) the scope of the presentation has focused on obtaining well-known results of Einstein s theory of general relativity particularly in connection with the correct determination of the perihelion motion of the planet Mercury (iv) finally, we have obtained a surprising relation with Godel s celebrated incompleteness theorem. 

Before ending this section on relativity theory, we reflect on a remark made by Lowdin [27] regarding the perihelion motion of Mercury. Describing a gravitational approach within the consmiction of special relativity, he demonstrated that the perihelion moved but that the effect was only half the correct value. The problem here is the fundamental inconsistency between the force-, momentum and the energy laws, while the discrepancy for so-called normal distances are almost impossible to observe directly since (1 - x(r)) 1. However using the present method to the classical constant of motion... 

The superiority of Einstein s over Newton s theory became manifest in 1915, when Einstein could for the first time explain an anomaly in the motion of the planet Mercuiy (advance of the perihelion), known obseiwationally since 1859. He also predicted that... 

Shorter time-scale changes, and more immediately relevant, are due to the astronomical motions of the Earth and sun. The Earth s annual orbit around the sun is slightly elliptical and the Earth-sun distance varies, leading to small changes in the available energy throughout the year. The current eccentricity of the orbit means that the Earth is closest to the sun (perihelion) in the January, the Southern Hemisphere summer (Northern Hemisphere winter) and furthest from... 

Fig. 7.—Orbit (rosette) of the electron about the nucleus, taking into account the relativistic variability of mass the motion is doubly periodic, the perihelion being displaced by the angle A(f> per revolution.

Since the motion consists of a libration of r, combined with a uniform rotation of the perihelion, the form of the orbit is that of a rosette (cf. fig. 12). 

The motion is therefore doubly degenerate, since for a given value of J the energy is independent of Ja (the total angular momentum) as well as of J. Not only the longitude of the node, but also the angular distance of the perihelion from the line of nodes, remains unaltered. We have only one quantum condition,... 

The stationary motions in a weak electric field are, however, essentially different from those in a spherically symmetrical field differing only slightly from a Coulomb field. In the latter (for which the separation variables are polar co-ordinates) the path is plane it is an ellipse with a slow rotation of the perihelion. In the former (separable in parabolic co-ordinates) it is likewise approximately an ellipse, but this ellipse performs a complicated motion in space. If then, in the limiting case of a pure Coulomb field, k or nf be introduced as second quantum number, altogether different motions would be obtained in the two cases. The degenerate action variable has therefore no significance for the quantisation. 

This motion of the perihelion, together with its accompanying phenomena, can best be studied by referring to the curve representing the motion in the (wa°, J2°)-plane (fig. 32). Its equation is, by (4),... 

The secular motions of the orbit under the influence of the electric field are thus as follows while the line of nodes revolves once, the perihelion of the orbital ellipse performs two oscillations about the meridian plane perpendicular to the line of nodes. For a transit through this meridian plane in one direction, the total momentum J2°/27t is a maximum and consequently the eccentricity is a minimum for a transit in the other direction the eccentricity is a maximum. Since the component Js°/2it of the angular momentum in the direction of the field remains constant, the inclination of the orbital plane oscillates with the same frequency as the eccentricity. It has its maximum or minimum value when the perihelion passes through the equilibrium position, and it assumes both its maximum and minimum value twice during one revolution of the line of nodes. The major axis remains constant during this oscillation of orbital plane and perihelion (since Jj0 remains constant) the eccentricity varies in such a way that the electrical centre of gravity always remains in the plane... 

An example will at once make clear what is meant. Let us take the relativistic Kepler motion, or, in other words, motion in an ellipse with a perihelion rotation. In general, the path fills a circular ring and, therefore, a two-dimensional region, densely everywhere. The boundaries for the libration of the radius vector are here concentric circles. 

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