Topological, Differential Geometry Methods and Modified Variational Approach for Calculation of the Propagation Time of a Signal, Emitted by a GPS-Satellite and Depending on the Full Set of 6 Kepler Parameters Parameters

TL;DR

This paper extends a geometric and variational method for calculating the propagation time of signals emitted by satellites, incorporating all six Kepler parameters and applying topological concepts like Morse functions.

Contribution

It introduces a novel formalism combining differential geometry, topology, and variational calculus to account for all Kepler parameters in signal propagation time calculations.

Findings

The formalism accounts for all six Kepler parameters.

Morse functions cannot be defined with respect to the omega angle.

The approach uses a quadratic functional in Kepler parameter differentials.

Abstract

Previously a mathematical approach has been developed for calculation of the propagation time of a signal, emitted by a moving along an elliptical orbit satellite, with account also for the General Relativity Theory (GRT) effects. The formalism was restricted to one dynamical parameter (the true anomaly or the eccentric anomaly angle). In this paper the aim is to extend the formalism to the case, when also the other five Kepler parameters will be changing.The following problem can be formulated: if two satellites move on two space-distributed orbits and they exchange signals, how can the propagation time be calculated? In this paper approaches from differential geometry and topology were implemented.The action functional for the propagation time is represented in the form of a quadratic functional in the differentials of the Kepler elements. The known mapping from celestial mechanics is used, when by means of a transformation the 6 Kepler parameters are mapped into the cartesian coordinates X, Y, Z. This is in fact a submersion of a manifold of 6 parameters into a manifold of 3 parameters. If a variational approach is applied with respect to a differential form in terms of the differentials of the Kepler parameters, the second variation will be different from zero and the Stokes theorem can be applied, provided that the second partial derivatives of the Cartesian coordinates with respect to the Kepler parameters are assumed to be different from zero. From topology viewpoint this requirement is equivalent to the existence of the s.c. Morse functions (non-degenerate at the critical points). In the given case it has been shown that Morse function cannot exist with respect to each one of the Kepler parameters- Morse function cannot be defined with respect to the omega angle.