Two null gravitational cones in the theory of GPS-intersatellite communications between two moving satellites. I. Physical and mathematical theory of the space-time interval and the geodesic distance on intersecting null cones

Dimitrov, Bogdan G.

doi:10.48550/arxiv.1712.01101

Cited by 2 publications

(7 citation statements)

References 45 publications

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“…This position depends on the time of motion of the satellite, but since it is equal to the propagation time, it is also an unknown variable. This problem can be solved if the formalism of two null cones, developed in [8] and [9] is applied. In fact, this inconvenience is typical also for the Shapiro delay formulae.…”

Section: Discussionmentioning

confidence: 99%

“…There is a second motivation for finding the propagation time for the various cases, related to the formalism of the two intersecting null cones with origins at the signal-emitting and signal receiving satellites, developed in the paper [8] (see also a shorter version in [9]). This formalism enables to find the propagation time of the signal, when both satellites are moving, meaning that the propagation time of the signal takes into account the motion of the second (signal-receiving) satellite during the time for propagation of the signal between the two satellites.…”

Section: Introductionmentioning

confidence: 99%

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Elliptic Integrals for Calculation of the Propagation Time of a Signal, Emitted and Received by Satellites on One Orbit

Dimitrov¹

2022

Preprint

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The propagation time of a signal, emitted by a moving along an elliptical orbit satellite from the GPS (or GLONASS) satellite confi gurations is a very important ingredient of the theory, based on the formalism of the null cone and accounting for the effects of the General Relativity Theory. For the case of satellites, orbiting along a plane elliptic orbit, it has been proved that the propagation time for the signal between the satellites is given by a combination of elliptic integrals of the first, second and third kind. For the more general case of satellites on a space-distributed elliptic orbit, the propagation time is expressed by higher (fourth) order elliptic integrals, which according to the standard theory can be expressed recurrently by means of lowerorder elliptic integrals. In the concrete case, the elliptic integrals of the second and the fourth order are expressed by means of a combination of irrational functions and the zero-order elliptic integral in the Legendre form. It has been proved that for the investigated case, second-order elliptic integrals can be expressed by elementary functions.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Elliptic Integrals for Calculation of the Propagation Time of a Signal, Emitted and Received by Satellites on One Orbit

Dimitrov¹

2022

Preprint

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“…If the above equation is integrated as a differential equation in full derivatives (see the details in the paper [13]), then the following symmetrical under interchange of the indices 1 and 2 expression can be derived…”

Section: Another Expression For the Variable Euclidean Distance From ...mentioning

confidence: 99%

“…This report has the purpose just to outline the basic topics and main conclusions in a more extensive and detailed theoretical research, performed in the paper [13]. In this paper, the investigation is restricted to intersatellite communications between satellites on one orbital plane.…”

Section: Introductionmentioning

confidence: 99%

“…The theory in the paper [13] is just the first step towards constructing a theory for propagation of signals between moving satellites on different, space-oriented orbits. The main motivation comes from the requirement that the Global Navigation Satellite System (GNS S ), consisting of 30 satellites and orbiting the Earth at a height of 23616 km, should be interoperable with the other two navigational systems GPS and GLONAS S [15].…”

Section: Introductionmentioning

confidence: 99%

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New mathematical models of GPS intersatellite communications in the gravitational field of the near-Earth space

Dimitrov¹

2019

AIP Conference Proceedings

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Several space missions such as GRACE, GRAIL, ACES and others rely on intersatellite communications (ISC) between two satellites at a large distance one from another. The main goal of the theory is to formulate all the navigation observables within the General Relativity Theory (GRT). The same approach should be applied also to the intersatellite GPS-communications (in perspective also between the GPS, GLONASS and Galileo satellite constellations). In this paper a theoretical approach has been developed for ISC between two satellites moving on (one-plane) elliptical orbits based on the introduction of two gravity null cones with origins at the emitting-signal and receiving-signal satellites. The two null cones account for the variable distance between the satellites during their uncorrelated motion. This intersection of the two null cones gives the space-time interval in GRT. Applying some theorems from higher algebra, it was proved that this space-time distance can become zero, consequently it can be also negative and positive. But in order to represent the geodesic distance travelled by the signal, the space-time interval has to be "compatible" with the Euclidean distance. So this "compatibility condition", conditionally called "condition for ISC", is the most important consequence of the theory. The other important consequence is that the geodesic distance turns out to be the space-time interval, but with account also of the "condition for ISC". The geodesic distance is proved to be greater than the Euclidean distancea result, entirely based on the "two null cones approach" and moreover, without any use of the Shapiro delay formulae. Application of the same higher algebra theorems shows that the geodesic distance cannot have any zeroes, in accord with being greater than the Euclidean distance. The theory also puts a restriction on the eccentric anomaly angle E=45.00251 [deg], which is surprisingly close to the angle of disposition of the satellites in the GLONASS satellite constellation (the Russian analogue of the American GPS) -8 satellites within one and the same plane equally spaced at 45 deg. The approach is the first step towards constructing a new, consistent relativistic theory of ISC between moving satellites on different space-distributed Kepler orbits, which is a much more complicated problem not being solved for the moment.

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Two null gravitational cones in the theory of GPS-intersatellite communications between two moving satellites. I. Physical and mathematical theory of the space-time interval and the geodesic distance on intersecting null cones

Abstract: Two null gravitational cones in the theory of GPS-intersatellite communications between two moving satellites. I. Physical and mathematical theory of the space-time interval and the geodesic distance on intersecting null cones.

Cited by 2 publications

References 45 publications

Elliptic Integrals for Calculation of the Propagation Time of a Signal, Emitted and Received by Satellites on One Orbit

Elliptic Integrals for Calculation of the Propagation Time of a Signal, Emitted and Received by Satellites on One Orbit

New mathematical models of GPS intersatellite communications in the gravitational field of the near-Earth space

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