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

In preceding publications a mathematical approach has been developed for calculation of the propagation time of a signal, emitted by a moving along an elliptical orbit satellite and accounting also for the General Relativity Theory (GRT) Effects. So far, the formalism has been restricted to one dyna...

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Bibliographic Details
Published in:Journal of physics. Conference series Vol. 2910; no. 1; pp. 12001 - 12039
Main Author: Dimitrov, Bogdan G.
Format: Journal Article
Language:English
Published: Bristol IOP Publishing 01.12.2024
Subjects:
Cartesian coordinates
Celestial mechanics
Critical point
Differential geometry
Elliptical orbits
Extremum values
Formalism
Geometry
Manifolds (mathematics)
Mathematical analysis
Orbit calculation
Orbits
Parameter modification
Propagation
Quadratic forms
Relativity
Satellite navigation systems
Satellites
Stokes theorem (vector calculus)
Straight lines
Topology
ISSN:1742-6588, 1742-6596
Online Access:Get full text
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Summary:In preceding publications a mathematical approach has been developed for calculation of the propagation time of a signal, emitted by a moving along an elliptical orbit satellite and accounting also for the General Relativity Theory (GRT) Effects. So far, the formalism has been 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 and thus, the following important problem can be stated: if two satellites move on two space-distributed orbits and they exchange signals, how can the propagation time be calculated? This paper requires the implementation of differential geometry and topological methods. In this approach, the action functional for the propagation time is represented in the form of a quadratic functional in the differentials of the Kepler elements, consequently the problem is related to the first and second quadratic forms from differential geometry. Such an approach has a clear advantage, because if the functional is written in terms of Cartesian coordinates X, Y, Z, the extremum value after the application of the variational principle is shown to be the straight line - a result, known from differential geometry, but not applicable to the current problem of signal exchange between satellites on different orbits. So 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 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.
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ISSN:1742-6588
1742-6596
DOI:10.1088/1742-6596/2910/1/012001