From emission to inertial coordinates: an analytical approach

In the framework of relativistic positioning systems in Minkowski space-time, the determination of the inertial coordinates of a user involves the bifurcation problem (which is the indeterminate location of a pair of different events receiving the same emission coordinates). To solve it, in addition to the user emission coordinates and the emitter positions in inertial coordinates, it may happen that the user needs to know independently the orientation of its emission coordinates. Assuming that the user may observe the relative positions of the four emitters on its celestial sphere, an observational rule to determine this orientation is presented. The bifurcation problem is thus solved by applying this observational rule, and consequently, all of the parameters in the general expression of the coordinate transformation from emission coordinates to inertial ones may be computed from the data received by the user of the relativistic positioning system.

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“…Some preliminary results of this work were presented at the Spanish Relativity meeting ERE-2010 [16].…”

Section: A Outline Of the Papermentioning

confidence: 99%

“…For this purpose, one has to consider the barycenter of the emitters as the convenient reference event rather than one of the emitters. This issue will be addressed elsewhere [26], in connection with the symmetric formulation of the location problem in flat space-time.…”

Section: A Covariant Expression Of the Solutionmentioning

confidence: 99%

Positioning systems in Minkowski space-time: Bifurcation problem and observational data

2012

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“…The inertial coordinates (user position) may be found by using the satellite world lines and the emission proper times, excepting some cases in which the emission coordinates are compatible with two user positions (bifurcation); in these cases, a criterion -based on additional data-is necessary to choose the true position (Schmidt, 1972;Abel and Chaffee, 1991;Chaffee and Abel, 1994;Grafarend and Shan, 1996;Coll et al, 2011Coll et al, , 2012Puchades and Sáez, 2012). Bifurcation does not play a role in this paper.…”

Section: Emission and Inertial Coordinates In The 0-order Rpsmentioning

confidence: 99%

Approaches to relativistic positioning around Earth and error estimations

Puchades

Sáez

2016

Advances in Space Research

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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.

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“…See Schmidt (1972); Abel & Chaffee (1991); Chaffee & Abel (1994); Grafarend & Shan (1996). In this case, additional information is necessary to get the true position (Coll, Ferrando & Morales-Lladosa 2011Puchades & Sáez 2012).…”

Section: Introductionmentioning

confidence: 99%

Relativistic positioning: errors due to uncertainties in the satellite world lines

Puchades

Sáez

2014

Astrophys Space Sci

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Global navigation satellite systems use appropriate satellite constellations to get the coordinates of an user -close to Earth-in an almost inertial reference system. We have simulated both GPS and GALILEO constellations. Uncertainties in the satellite world lines lead to dominant positioning errors. In this paper, a detailed 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. Our study is based on the Jacobian (J) of the transformation giving the emission coordinates in terms of the inertial ones. Around points of vanishing J, positioning errors are too large. We show that, for any 4-tuple of satellites, the points with J = 0 are located at distances, D, from the Earth centre greater than about 2R/3, where R is the radius of the satellite orbits which are assumed to be circumferences. Our results strongly suggest that, for Ddistances greater than 2R/3 and smaller than 10 5 km, a rather good positioning may be achieved by using appropriate satellite 4-tuples without J = 0 points located in the user vicinity. The way to find these 4-tuples is discussed for arbitrary users with D < 10 5 km and, then, preliminary considerations about satellite navigation at D < 10 5 km are presented. Future work on the subject of space navigation -based on appropriate simulations-is in progress.

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From emission to inertial coordinates: an analytical approach

Cited by 4 publications

References 6 publications

Positioning systems in Minkowski space-time: Bifurcation problem and observational data

Positioning systems in Minkowski space-time: Bifurcation problem and observational data

Approaches to relativistic positioning around Earth and error estimations

Relativistic positioning: errors due to uncertainties in the satellite world lines

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