Algebraic Geometry and Elliptic Integrals Approach for Calculation of the Propagation Time of a Signal Between GPS Satellites with Account of General Relativity Theory

Abstract

The approach for calculating the propagation time of a signal between GPS satellites will be summarized, based on the proposed new theoretical approach in several previous publications, as well as the perspectives for future development of the theory. Topics include: 1. Basic notions of inter-satellite communications. 2. Shapiro delay formulae in General Relativity Theory - basic formalism and the necessity to extend the formalism by taking into account the satellite motion on a plane elliptic or space-distributed elliptic orbit. 3. Basic facts about the disturbed motion in celestial mechanics and the necessity to incorporate it in the theory of inter-satellite communications, accounting for General Relativity Effects. 4. Propagation time of a signal, emitted by a satellite on a plane and also space-distributed elliptical orbit in terms of zero-order elliptic integrals and respectively of higher order integrals. Proof of the real-valuedness of the propagation time for all cases as one of the criteria for the correctness of the theoretical approach. 5. New analytical algorithms for calculation of zero-order elliptic integrals in the Legendre form. Relation to two representations in the Weierstrass form. 6. The new formalism of intersecting four-dimensional null cones and the resulting physical notions of the (intersecting)) space-time interval (with the property of being positive, negative or equal to zero) and the (intersecting) geodesic distance (being only positive, because is related to the distance, travelled by light or radio signals). Proof of these properties in the general case and in some partial cases. New numerical estimate E lim > 45.002510943228 [deg], above which the space-time interval is positive and thus inter-satellite communications between satellites on one plane elliptical orbit are possible. The angular distance of 45[ deg] is typical for the disposition of 8 satellites on one orbit in the Russian satellite constellation GLON ASS, so it might be claimed that such a configuration is favourable from the point of view of inter-satellite communications (with account of GRT effects).