Approaches to relativistic positioning around Earth and error estimations

TL;DR

This paper compares errors in relativistic positioning around Earth caused by satellite uncertainties and photon path assumptions, finding that simplified Minkowski space-time models are sufficiently accurate for practical navigation.

Contribution

It introduces a comparison between U-errors and S-errors in satellite-based relativistic positioning, validating the use of Minkowski space-time assumptions for Earth-centered navigation.

Findings

S-errors are smaller than U-errors under realistic conditions.

Simplified Minkowski space-time assumptions are adequate for Earth-centered positioning.

More complex models are unnecessary for typical navigation accuracy.

Abstract

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.