Relativistic Positioning System in perturbed spacetime

Abstract

Abstract. We present a variant of a Global Navigation Satellite System called a Relativistic Positioning System (RPS), which is based on emission coordinates. We modelled the RPS dynamics in a space-time around Earth, described by a perturbed Schwarzschild metric, where we included the perturbations due to Earth multipoles (up to the 6th), the Moon, the Sun, Venus, Jupiter, solid tide, ocean tide, and Kerr rotation effect. The exchange of signals between the satellites and a user was calculated using a ray-tracing method in the Schwarzschild space-time. We find that positioning in a perturbed space-time is feasible and is highly accurate already with standard numerical procedures: the positioning algorithms used to transform between the emission and the Schwarzschild coordinates of the user are very accurate and time efficient -on a laptop it takes 0.04 s to determine the user's spatial and time coordinates with a relative accuracy of 10 −28 − 10 −26 and 10 −32 − 10 −30 , respectively.

Chronometric Geodesy: Methods and Applications

“…One drawback of harmonic frames is that the harmonic gauge condition does not admit rigidly rotating frames [51, chapter 8]. Other recent approaches are based on a perturbed Schwarzschild metric [52], or on the Kerr metric [53] in the different context of a slowly rotating astronomical object. Following the pioneering works, a set of Resolutions was adopted at the IAU General Assembly in Manchester in the year 2000 [54]: We summarize here very briefly these resolutions.…”

“…Moreover, the values are nearly uncorrelated over longer distances, with a correlation of less than 10 % beyond a distance of about 180 km [81]. Aiming at the determination of the absolute gravity potential W according to equations (51) or (52), which is the main advantage of the GNSS/geoid technique over the geometric levelling approach, both the uncertainties of GNSS and the quasigeoid have to be considered. Assuming a standard deviation of 1.9 cm for the quasigeoid heights and 1 cm for the GNSS ellipsoidal heights without correlations between both quantities, a standard deviation of 2.2 cm is finally obtained (in terms of heights) for the absolute potential values based on the GNSS/geoid approach.…”

“…(43)) and the GNSS/geoid approach (eqs. (51) or (52)). The results from both approaches are provided in the form of geopotential numbers according to equation (39) with…”

Theoretical Tools for Relativistic Gravimetry, Gradiometry and Chronometric Geodesy and Application to a Parameterized Post-Newtonian Metric

“…Explicit expressions of the metric components in these coordinate systems are given in [38]. Other approaches exist in this context based on generalized Fermi coordinates [39][40][41], or a perturbed Schwarzschild metric [42]. In the different context of a slowly-rotating astronomical object, the Kerr metric is used in [35].…”

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

“…They claimed that their approach was more satisfactory than the previous ones especially with regard to its consistency, completeness and flexibility. In Kostić et al (2015) a model of RPS is presented in a more realistic spacetime near the Earth with all important gravitational effects: Earth multipoles up to 6th order, the Moon, the Sun, Jupiter, Venus, solid and ocean tides, and Kerr effect. A recent approach was proposed by Roh et al (2016), Roh (2018).…”