Relativistic positioning: including the influence of the gravitational action of the Sun and the Moon and the Earth’s oblateness on Galileo satellites

Abstract

Uncertainties in the satellite world lines lead to dominant positioning errors. In the present work, using the approach presented in Puchades and Sáez (Astrophys. Space Sci. 352, 307–320, 2014), a new analysis of these errors is developed inside a great region surrounding Earth. This analysis is performed in the framework of the so-called Relativistic Positioning Systems (RPS). Schwarzschild metric is used to describe the satellite orbits corresponding to the Galileo Satellites Constellation. Those orbits are circular with the Earth as their centre. They are defined as the nominal orbits. The satellite orbits are not circular due to the perturbations they have and to achieve a more realistic description such perturbations need to be taken into account. In Puchades and Sáez (Astrophys. Space Sci. 352, 307–320, 2014) perturbations of the nominal orbits were statistically simulated. Using the formula from Coll et al. (Class. Quantum Gravity. 27, 065013, 2010) a user location is determined with the four satellites proper times that the user receives and with the satellite world lines. This formula can be used with any satellite description, although photons need to travel in a Minkowskian space-time. For our purposes, the computation of the photon geodesics in Minkowski space-time is sufficient as demonstrated in Puchades and Sáez (Adv. Space Res. 57, 499–508, 2016). The difference of the user position determined with the nominal and the perturbed satellite orbits is computed. This difference is defined as the U-error. Now we compute the perturbed orbits of the satellites considering a metric that takes into account the gravitational effects of the Earth, the Moon and the Sun and also the Earth oblateness. A study of the satellite orbits in this new metric is first introduced. Then we compute the U-errors comparing the positions given with the Schwarzschild metric and the metric introduced here. A Runge-Kutta method is used to solve the satellite geodesic equations. Some improvements in the computation of the U-errors using both metrics are introduced with respect to our previous works. Conclusions and perspectives are also presented.

Study Review of the Speed of Light in Space-Time for STEM Student

“…This means that the satellite's clock will appear to run slower to an observer on the surface of the more massive object, compared to the clock of an observer who is farther away from the gravitational field. This effect is due to the fact that the satellite is constantly accelerating as it moves in its circular orbit, and this acceleration results in a difference in the passage of time (Colmenero et al, 2021). The students may have proposed to analyze this case because they are interested in understanding how gravitational time dilation works in more detail, and how it might affect the behavior of objects in circular orbits.…”