Approaches to relativistic positioning around Earth and error estimations

In the context of relativistic positioning, the coordinates of a given user may be calculated by using suitable information broadcast by a 4-tuple of satellites. Our 4-tuples belong to the Galileo constellation. Recently, we estimated the positioning errors due to uncertainties in the satellite world lines (U-errors). A distribution of U-errors was obtained, at various times, in a set of points covering a large region surrounding Earth. Here, the positioning errors associated to the simplifying assumption that photons move in Minkowski space–time (S-errors) are estimated and compared with the U-errors. Both errors have been calculated for the same points and times to make comparisons possible. For a certain realistic modeling of the world line uncertainties, the estimated S-errors have proved to be smaller than the U-errors, which shows that the approach based on the assumption that the Earth’s gravitational field produces negligible effects on photons may be used in a large region surrounding Earth. The applicability of this approach – which simplifies numerical calculations – to positioning problems, and the usefulness of our S-error maps, are pointed out. A better approach, based on the assumption that photons move in the Schwarzschild space–time governed by an idealized Earth, is also analyzed. More accurate descriptions of photon propagation involving non symmetric space–time structures are not necessary for ordinary positioning and spacecraft navigation around Earth.