File Acta Futura Acta. Futura. Relativistic Positioning Systems

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 system, 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-  µν  0 0 −h0 csc θ∂φ h0 sin θ∂θ dependent background with the Schwarzschild metric   (0)  0 0 −h1 csc θ∂φ h1 sin θ∂θ  m (6) gµν and denote metric perturbations with hµν. Be- Y ,  ⋆ ⋆ 0 0  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) =  µν  (0) gµν = gµν + hµν . (1) H0χ H1 0 0  ⋆ H χ−1 0 0  (7)  2  Ym , Because we are interested in the space-time outside  0 0 r2K 0  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 coordinates (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] . In the RWZ formalism, the metric perturbation . (8) diag(Hχ, Hχ−1, r2K, r2K sin2 θ)Ym hµν is expanded into a series of independent tensor har- n monics, a tensor analog to spherical harmonic functions, Inserting it in (4) gives equations for H and K. We find labeled by indices n (degree), m (order), and parity: odd the solution for H to be: or even. By adopting the notation from [14], the gen- ( ) ( ) (0) rs n+1 (0) rs eral expansion of the metric perturbation hµν can be Pn r r Rn r H(r) = Anm n + Bnm , (9) written as r (r − rs) r − rs ∑∞ ∑n (0) (0) nm (o) nm (e) where the functions Pn and Rn are Gaussian hyper- hµν = (h ) + (h ) , (5) µν µν geometric functions 2F1 [1] (for more details see[11]). n=2 m=−n e solution for function K is determined by H (see nm (o) nm (e) [11]) and can be written as where the expansion terms (hµν ) and (hµν ) are ( ) (1) rs ( ) the odd-parity and the even-parity metric functions (or Pn r K(r) = A r + B rnR(1) s , (10) modes), respectively. We find it most convenient to nm rn+1 nm n r

DOI: 10.2420/AF07.2013.79 81 Acta Futura 7 (2013) / 79-85 A. Gomboc et al.

(1) (1) (0) where functions Pn and Rn can be expressed via Pn Odd-parity contributions (0) and Rn [11]. In case of time-independent perturbations, the odd In our case of satellites in orbit around Earth, time- nm (o) metric functions (h ) in (6) have h1 = 0 [15] and independent perturbations are due to the Earth’s multi- µν can be written with a single function h0 as poles (for the time being, we neglect Earth’s rotation and tides). In order to determine the constants Anm and nm (o) m (hµν ) = − h0 csc θ Yn ,φ(δ0,µδ2,ν + δ2,µδ0,ν) Bnm in (9) and (10), we compare equation (9) with its + sin m ( + ) . Newtonian counterpart, i.e., the gravitational potential h0 θ Yn ,θ δ0,µδ3,ν δ3,µδ0,ν Φ. For a non-rotating object of mass M the potential Φ (15) can be expanded into a series of multipole contributions e solution for h is: (eq. 3.61 in [12]): 0 ( ) (2) rs ( ) Pn r GM h (r) = α r + β rn+1R(2) s , Φ = 0 nm n nm n (16) r r r ∑ (11) ⊕ −n−1 ⊖ n m (2) (2) + (Mnmr + Mnmr )Yn , where functions Pn and Rn are Gaussian hypergeo- nm metric functions 2F1 (for more details see[11]). ⊕ ⊖ To determine the constants α and β we note where M and M are time-independent spher- nm nm nm nm ∑ that off-diagonal terms in the metric tensor are associ- ical multipole momenta and notation ≡ ∑∞ ∑ nm ated with frame-dragging effects. In our case frame- n is used. e first term in the sum n=2 m=−n dragging effects from ’external’ objects (the Sun, the describes the gravitational potential of the perturbing Moon, other planets) are negligible. Consequently, we sources positioned within the radius r, while the sec- set β = 0. ond term corresponds to those outside r. Comparing nm We do take into account the frame-dragging due to (9) with (11), we notice the same behavior (i.e., the su- the Earth’s rotation. To determine α , we notice that perposition of r+n and r−n−1 functional dependence) nm for n = 1 and m = 0 the corresponding h matches the in the perturbative part of (11) and it is evident that the 0 weak field and slow rotation approximation of the Kerr coefficients A and B are related to the multipole nm nm metric: if r ≫ r and Earth’s angular parameter a ≪ 1, momenta. e relation between both is found from the s √ then for α = ar 4π/3 it follows weak field approximation 10 s √ r 4π c2 h (r) = a s , (1 + g ) ∼ Φ . (12) 0 (17) 2 00 r 3 where we keep only the terms linear in a. By inserting For higher multipoles (n > 1), it turns out that their ∑ dependence on a is not linear [10]. erefore, the only g = g(0) + (hnm)(e) 00 00 00 multipole we include in the odd-parity metric function ( nm ) ∑ (13) is the monopole, i.e., the one belonging to the linear (in a) part of the Kerr effect. = χ −1 + Hnm nm 2.2 Time-dependent metric perturbations into the above relation together with the Newtonian po- Due to the Earth’s rotation its multipoles vary periodi- tential (11), we find that in the weak field limit Anm and cally. Earth’s solid and ocean tides introduce additional Bnm are asymptotically related to Newtonian spherical multipole momenta M⊕ and M⊖ as time dependency in its multipoles (additional variability nm nm with different frequency, phase and varying amplitude,

2 ⊕ 2 ⊖ depending on the position of the Moon and the Sun). A ∼ M and B ∼ M . (14) nm c2 nm nm c2 nm In addition, the gravitational inﬂuence of other celestial bodies introduces time dependent perturbations to ⊕ ⊖ Note that for ﬁnite c, Mnm and Mnm only approxi- the space-time around Earth, because their relative po- mate Anm and Bnm. sitions with respect to Earth change with time. ese

82 DOI: 10.2420/AF07.2013.79 Relativistic Positioning Systems and Gravitational Perturbations perturbations can be expanded in a series of multipoles e first terms in the expressions for H and K are already e e and treated with the same procedure as the Earth’s mul- known: H(0) is given in (9) and K(0) in (10), where in- tipoles. stead of (14) we use: We therefore consider time dependent metric perturbations for the case of perturbations oscillating slowly ∼ 2 f⊕ ∼ 2 f⊖ Anm Mnm and Bnm Mnm . (23) with angular velocities, which are smaller or of the same c2 c2 order of magnitude as the angular velocity of Earth. For H˜ we find that All angular velocities are defined with respect to the 1 ( ) Schwarzschild time t. −n+1 (3) rs e (0) r Pn r H1 (r) = Anm rs(r − rs) Even-parity contributions ( ) (24) n+2 (3) rs r Rn r Even parity modes are connected to the Newtonian + Bnm , r (r − r ) gravitational potential Φ, which in the case of time de- s s pendent multipoles can be written as: (3) (3) rs where functions Pn and Rn are given as a series in r GM [11]. Φ = Because k is very small for all time-dependent per- ∑r ⊕ ⊖ (18) turbations considered, we neglect all higher than lead- + (M (T)r−n−1 + M (T)rn)Ym , nm nm n ing terms in the expansion (22) and use the following nm approximations: where T = ct, or alternatively in frequency domain: ∫ ∞ ∑ ∫ ∞ ≈ ikT e (0) GM H(T, r) dk e H (k, r) , (25) Φ = + dk eikT × −∞ ∫ ∞ r −∞ [ nm ] (19) ≈ ikT e (0) H1(T, r) dk ikrse H1 (k, r) , (26) f⊕ − −1 f⊖ M (k)r n + M (k)rn Ym , −∞ nm nm n ∫ ∞ i e(0) K(T, r) ≈ dk e kT K (k, r) . (27) f⊕ f⊖ ∞ where k is the wavenumber and Mnm, Mnm are the − Fourier transforms of time dependent multipoles: A metric perturbation expressed with these functions ∫ ∞ fu 1 −ikT u is accurate up to the linear order in frequency. Since Mnm(k) = dT e Mnm(T) , (20) higher order perturbations naturally give rise to contri- 2π −∞ butions with higher orders of frequencies, our approxi- where u = ⊕, ⊖. mation of a perturbation is consistently linear, i.e., it is Each time dependent multipole generates a time- linear in frequencies and in the order of perturbation. nm (e) dependent even metric perturbation (hµν ) . Func- tions H, H1, and K determining the modes can be ex- Odd-parity contributions pressed with their Fourier transforms as For odd-parity solutions we use the same notation as in (H(T), H1(T), K(T)) = (21) and we find that the asymptotic behaviour of solu- ∫ ∞ tions h˜ 0 and h˜ 1 is not flat: ikT e e e (21) dk e (H(k), H1(k), K(k)) −∞ h˜ 1(r) ≍ r sin(kr + ϕ) (28) ≪ Because in our case k is very small (k 1/r), we h˜ 0(r) ≍ r cos(kr + ϕ) (29) solve differential equations for H, H1, and K perturba- e e e tivelly in k. We assume that H, H1, and K are smooth ese solutions are therefore not relevant in our case. functions of k and write them as a power series of di- mensionless κ = krs: 3 Metric around Earth ∑∞ e e e ∼ 2i e (i) e (i) e(i) Finally, we can write the metric perturbation hµν, (H, H1, K) κ (H , iκH1 , K ) , (22) i=0 which in (5) was expressed as a series of normal modes

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⊖ nm (o) nm (e) (hµν ) and (hµν ) . Based on the positions of the where Mnm are summed multipoles of other celestial sources of perturbations, these modes can be grouped bodies (in our case the Sun, the Moon, Jupiter and into two terms: Venus). ⊕ ⊖ e first order approximations of the metric pertur- hµν = hµν + hµν . (30) bations given by (32) and (33) are fully determined by ⊕ ⊖ ⊕ multipole momenta M , M , Kerr parameter a, e term hµν represents the Earth’s time dependent nm nm (i) (i) (exterior) multipoles and the frame-dragging effect of and functions Pn and Rn (for details see [11]). Earth. e former arise from the shape of the Earth, which changes with time due to rotation and tidal forces. 4 Conclusions For the latter, Kerr effect, there is no non-relativistic counterpart. e aim of our project is to model the Galileo GNSS ⊖ e term hµν represents the time dependent (in- in general relativity including all relevant gravitational terior) multipoles of other celestial bodies. is term perturbations and to test to what level can this new ap- arises from the perturbative effect of other planets, the proach improve the accuracy and stability of the Galileo Moon, and the Sun, whose positions relative to Earth’s GNSS reference frame. change with time. eir frame-dragging effect is ne- In this contribution, we show how we can include glected. gravitational perturbations with linear perturbation the- To simplify expressions, we introduce the normalized ory on a Schwarzschild background and present explicit complex multipoles (u = ⊕, ⊖): expressions (32) and (33) for the perturbative metric. In the next steps, we will use this metric in the per- 2 u u turbative Hamiltonian formalism to obtain time deriva- Mnm := 2 Mnm . (31) c tives of zeroth order constants of motion. With the Metric perturbation due to Earth’s multipoles and ro- derivatives known, we will be able to determine the time tation can be written as evolution of these slowly changing constants of motion ∑ ⊕ and apply them to analytical solutions for Schwarzschild ⊕ m× [hµν] = MnmYn geodesics to obtain satellites’ orbits in perturbed space- ( nm ) time. Once the orbits are known, we can use them to (0) (0) (1) (1) 2 Pn Pn Pn Pn sin θ do the relativistic positioning, as well as build the ABC diag n+1 , n−1 2 , n−1 , n−1 r r (r − rs) r r in the perturbed space-time around Earth, using only inter-link communication between GNSS satellites. ∑ (3) ⊕ P + M Ym n (δ δ + δ δ ) nm,T n rn−1(r − r ) µ,1 ν,0 ν,1 µ,0 nm s Acknowledgements rs 2 − a⊕ sin θ(δ δ + δ δ ) , r µ,3 ν,0 ν,3 µ,0 AG, MH and UK acknowledge the financial support by (32) ESA Slovenian PECS project Relativistic Global Navi-

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