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Acta Futura 7 (2013) 79-85 Acta DOI: 10.2420/AF07.2013.79 Futura Relativistic Positioning Systems and Gravitational Perturbations A G,* Faculty of Mathematics and Physics, University of Ljubljana, Jadranska ulica 19, 1000 Ljubljana, Slovenia Centre of Excellence SPACE-SI, Aškerčeva cesta 12, 1000 Ljubljana, Slovenia U K, M H, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska ulica 19, 1000 Ljubljana, Slovenia S C, ESA - Advanced Concepts Team, ESTEC, Keplerlaan 1, Postbus 299, 2200 AG Noordwijk, Netherlands P D LNE-SYRTE, Observatoire de Paris, CNRS and UPMC ; 61 avenue de l’Observatoire, 75014 Paris, France Abstract. In order to deliver a high accuracy and time via the time difference between the emission relativistic positioning system, several gravitational and the reception of the signal. However, due to the in- perturbations need to be taken into account. We ertial reference frames and curvature of the space-time therefore consider a system of satellites, such as in the vicinity of Earth, space and time cannot be con- the Galileo system, in a space-time described by a sidered as absolute. In fact, general relativistic effects are background Schwarzschild metric and small grav- far from being negligible [2]. Current GNSS deal with itational perturbations due to the Earth’s rotation, this problem by adding general relativistic corrections multipoles and tides, and the gravity of the Moon, the Sun, and planets. We present the status of this to the level of the accuracy desired. An alternative, and work currently carried out in the ESA Slovenian more consistent, approach is to abandon the concept of PECS project Relativistic Global Navigation Sys- absolute space and time and describe a GNSS directly in tem, give the explicit expressions for the perturbed general relativity, i.e. to define a Relativistic Positioning metric, and briefly outline further steps. System (RPS) with the so-called emission coordinates [5, 16, 3, 4, 17]. A user of an RPS receives, at a given moment, four signals from four different satellites. It 1 Introduction is able to determine the proper time τ of each satellite at the moment of emission of these signals. en these Current Global Navigation Satellites Systems (GNSS), four proper times (τ1, τ2, τ3, τ4) constitute its emission such as the Global Positioning System and the Euro- coordinates. By receiving them at subsequent times, the pean Galileo system, are based on Newtonian concept receiver therefore knows its trajectory in the emission of absolute time and space. e signals from four satel- coordinates. lites are needed for a receiver to determine its position ere are several advantages of an RPS. Firstly, the *Corresponding author. E-mail: [email protected] emission coordinates are covariant quantities; they are 79 Acta Futura 7 (2013) / 79-85 A. Gomboc et al. independent of the observer (although dependent on the simulate their inter-satellite communication and use the set of satellites chosen and their trajectories). More- emission/reception of their proper times to determine over, if each satellite broadcasts its own and also receives their orbital parameters. Our work is divided in the fol- proper times of other satellites, the system of satellites lowing steps: is autonomous and constitutes a primary reference sys- tem, with no need to define a terrestrial reference frame. 1 Add first order gravitational perturbations to the erefore tracking of satellites with ground stations is Schwarzschild metric: find perturbation coeffi- necessary only to link an RPS to a terrestrial frame, al- cients describing all known gravitational perturba- though this link can also be obtained by placing several tions, i.e., due to Earth’s multipoles, tides and ro- receivers at the known terrestrial positions. ere is no tation; gravity of the Moon, the Sun, and planets need to synchronize satellite clocks to a time-scale real- (Venus and Jupiter). ized on the ground (like it is done today with GPS time, which is a realization of the TT Terrestrial Time). No relativistic corrections are necessary, as relativity is al- 2 Solve the perturbed geodesic equations: use ready included in the definition of the positioning sys- Hamiltonian formalism and perturbation theory to tem. obtain time evolution of zeroth order constants of To demonstrate feasibility, stability and accuracy of motion. Simulate satellites’ orbits. an RPS, two ESA Ariadna projects were carried out in 2010 and 2011 [6, 7, 8]. In these projects, the authors 3 Find accurate constants of motion: use solely inter- modelled an RPS in the idealized case of Schwarzschild satellite links over many orbital periods to deter- geometry and found that relativistic description of a mine satellites’ orbital parameters and study the satellite constellation with inter-satellite links provides stability and possible degeneracies between them. numerically accurate, stable and autonomous system. ey called such a system the Autonomous Basis of Co- 4 Refine values of gravitational perturbation coef- ordinates (ABC). By communicating their proper times ficients: use additional satellites and residual er- solely, two satellites can determine their orbits (i.e., their rors between orbit prediction and orbit determina- constants of motion). Any additional satellite would tion through inter-satellite communication to im- serve to increase the system’s accuracy. prove the accuracy of position determination and to probe the space-time, i.e., measure perturbations. Outline of the PECS Relativistic GNSS project and its Discuss possible scientific applications. goals Continuation of this work is presently being carried out Before starting detailed calculations it is useful to get in the ESA Slovenian PECS project Relativistic Global an estimation of the order of magnitude of various per- Navigation System (2011-2014). In this project, we aim turbations on a satellite orbit (as in e.g. [13, Fig. 3.1.]). to demonstrate that an RPS and the ABC concept are At GNSS altitudes of about 20.000 km above Earth, highly accurate and stable also if the space-time is not the most important gravitational perturbations are due purely spherically symmetric, but contains small gravita- to Earth’s multipoles, followed by the gravitational field tional perturbations due to the Earth’s multipoles, tides of the Moon and the Sun. Several orders of magnitude and rotation and gravitational influences of the Moon, smaller are perturbations due to Solar radiation pressure the Sun, Jupiter and Venus. In this work we use the and the Earth’s albedo (not considered in our project) same approach that was used in two Araidna projects and due to Earth’s tides. About one order of magnitude (description of satellites’ orbit in the ABC, emission co- smaller are relativistic effects and the gravitational in- ordinates, inter-satellite communication and recovery of fluence of Jupiter and Venus. Relativistic effects due to their orbits) and add to a Hamiltonian gravitational per- Earth’s rotation are about an order of magnitude smaller turbations. again. In brief: we start modelling of an RPS by writing Because our project is still in progress, we will present the perturbed Hamiltonian, corresponding equations of here only the results of the first part of the project: we motion and calculating orbits of satellites. en we de- will give the expressions for the perturbed metric around fine the system based on the ABC with four satellites, Earth. 80 DOI: 10.2420/AF07.2013.79 Relativistic Positioning Systems and Gravitational Perturbations 2 Metric perturbations in the work in the gauge from [15], where the odd parity met- Schwarzschild background ric functions are (hnm)(o) = We describe the spherically symmetric and time in- 2 µν 3 0 0 -h0 csc θ∂' h0 sin θ∂θ dependent background with the Schwarzschild metric 6 7 (0) 6 0 0 -h1 csc θ∂' h1 sin θ∂θ 7 m (6) gµν and denote metric perturbations with hµν. Be- Y , 4 ?? 0 0 5 n cause gravitational perturbations are several orders of ?? 0 0 magnitude smaller than the Earth’s gravitational GM (0) term (hµν ≪ gµν), we use linear perturbation theory and for even parity, the metric functions are and write the perturbed metric as (hnm)(e) = 2 µν 3 (0) gµν = gµν + hµν . (1) H0χ H1 0 0 6 ? H χ-1 0 0 7 (7) 6 2 7 Ym , Because we are interested in the space-time outside 4 0 0 r2K 0 5 n Earth, the perturbed metric must satisfy the Einstein 0 0 0 r2K sin2 θ equation for vacuum: where ? indicates the symmetric part of the tensor, χ = 2 1 1-rs=r, and rs = 2GM=c is the Schwarzschild radius. Rµν - gµνR = 0 (2) 2 Expressions hi, Hi, and K depend on Schwarzschild co- ordinates (t, r) and indices (n, m), which are omitted in where the expressions for clarity. It is shown in [15, 18] that in (0) Rµν = Rµν + δRµν (3) vacuum H0 = H2, therefore, both functions are marked with H. (0) is the Ricci tensor (symbol denotes unperturbed After inserting (6) and (7) into (4), we obtain a quantities and δ perturbations). e Einstein equation number of partial differential equations. We find and becomes: write their solutions for time-independent and time- α α α α dependent perturbations, separately. hα ;µν - hµ ;να - hν ;µα + hµν; α (0) λ α λ α (0) + gµν(hα ; λ - hλ ;α )- hµνR (4) 2.1 Time-independent metric perturbations (0) (0)λσ + gµνhλσR = 0 Even-parity contributions where a semi-colon (;) denotes covariant derivative, cal- In the case of time-independent perturbations of even (0) parity H1 = 0 [15]. It follows that the even metric mode culated with respect to the unperturbed metric gµν. To find solutions of these equations for vacuum we (7) is diagonal: use the Regge-Wheeler-Zerilli (RWZ) framework [15, nm (e) (hµν ) = 18] .