Numerical modeling of a Global Navigation Satellite System in a general relativistic framework |
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Authors: | Pacôme Delva Uroš Kosti? Andrej ?ade? |
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Institution: | 1. ESA Advanced Concepts Team, ESTEC, DG-PI, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands;2. Department of Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia |
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Abstract: | In this article we model a Global Navigation Satellite System (GNSS) in a Schwarzschild space–time, as a first approximation of the relativistic geometry around the Earth. The closed time-like and scattering light-like geodesics are obtained analytically, describing respectively trajectories of satellites and electromagnetic signals. We implement an algorithm to calculate Schwarzschild coordinates of a GNSS user who receives proper times sent by four satellites, knowing their orbital parameters; the inverse procedure is implemented to check for consistency. The constellation of satellites therefore realizes a geocentric inertial reference system with no a priori realization of a terrestrial reference frame. We perform a simulation of position determination and show that the determination of the four coordinates with a 25–32 digit accuracy takes only around 60 ms. Effects of non-gravitational perturbations on positioning errors are assessed, and methods to reduce them are sketched. In particular, inter-links between satellites could greatly enhance stability and accuracy of the positioning system. Effects of gravitational perturbations are omitted in this paper in order to make a clearer comparison between the relativistic and non-relativistic scheme, but they will be included in subsequent work. We believe that the final algorithm will be a serious alternative to the usual post-Newtonian scheme. |
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Keywords: | General relativity Schwarzschild space&ndash time Relativistic positioning system GNSS |
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