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

Abstract

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.

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

“…Unlike the case about the propagation times, the case about the atomic times is more complicated from a conceptual and technical point of view, because the defining formulae for the atomic time (see formulaes (43) and ( 44)) contains the Geocentric Coordinate Time (TCG), which is not a directly measurable quantity and by definition only enters the corresponding equations. However, by using two different representations for the atomic time, depending respectively on the propagation time and on the celestial time, it has been proved that for the given metric of the near-Earth space (22), the TCG can be identified with the celestial time (determined from the Kepler equation in celestial mechanics). This important proof in fact (contained in the derived formulaes (98)-( 100) ) makes possible the analytical and numerical calculation of the change of atomic time under the transportation of the atomic clock.…”

“…, in view of the identification of the geocentric time with the celestial time, which shall be proved for the case of the diagonal metric (22). If equations ( 47) and ( 48) are treated together with equation ( 25), then it will turn out that the first atomic time will depend on the eccentric anomaly angle E 1 because τ 1 from (47) can be expressed as…”

“…A more concise exposition of the new approach in this paper, concerning the space-time interval, the "condition for intersatellite communications" and the geodesic distance for the case of signal transmission between satellites, moving in a two-dimensional plane, can be found in the conference proceedings [22].…”

“…The method is developed in Section V B. The first differential is related to the expression for the given diagonal metric (22) and the second differential -to the expression (164) for the square of the differential of the formulae for the Euclidean distance. This method turned out to be suitable for the particular case of two-dimensional plane parametrization of the orbit, but could be developed also for the general case of space-distributed orbits.…”

“…where expressions ( 68), ( 62) and ( 63) have been used. For the diagonal metric (22), the above formulae can be written as…”