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Relativistic Modelling for Accurate Time Transfer via Optical Inter-Satellite Links

Relativistic Modelling for Accurate Time Transfer via Optical Inter-Satellite Links The DLR Institute for Communication and Navigation is currently working on a new GNSS architecture that enables accurate autonomous inter-satellite synchronization at picosecond-level. Synchronization is achieved via time transfer techniques enabled by optical inter-satellite links (OISLs), paving the way for a system in which space (orbits) and time (synchroniza- tion) can be effectively separated, leading to a high level of synchronization throughout the constellation, which in turn greatly improves accurate orbit determination. This is possible provided that relativistic effects are adequately taken into account. This work focuses on a two-way time transfer scheme based on the exchange of time stamps via optical signals, which allows the synchronization of a GNSS satellite system with respect to a defined coordinate time with picosecond-level accuracy. We analyse the impact of relativistic effects in clock offset estimation between optically linked clocks: results show that to achieve synchronization at this level of accuracy it is necessary to account for terrestrial geopotential harmonics up to the third order while the gravitational influence of additional celestial bodies can be neglected. Relativistic delays in the propagation of electromagnetic waves through spacetime are also evaluated. It is shown that for a two-way synchronization method, the Euclidean expression for the propagation of light is sufficient to achieve picosecond synchronization, provided m-level orbit determination of both satellites is available, and the hardware delays are well calibrated to the targeted accu- racy. Also, we show how to practically achieve autonomous synchronization via a sequence of pair-wise synchronizations across all satellites of the constellation. Keywords Time transfer · Proper time rate · Intersatellite links · Relativistic propagation 1 Introduction c t + t − t =  −  , j = 1, 2, 3, 4 (1) j j Position determination and time transfer in GNSS rely on a Having four independent equations, one can solve for the simple model. Consider four synchronized clocks located at four unknowns , t . Such a solution is also called a “navi- positions  , j = 1, 2, 3, 4 . The satellites transmit electromag- gation solution”. In principle the solution can be computed netic pulses at times t respectively. Suppose that these four with information from any number of satellites greater than signals are received at position  at instant t. The receiver or equal to four. Timing errors of 1 ns will lead to position- records the reception event happening at t + t where t is ing errors in the order of 30 cm. It is obvious that in order the offset of the receiver clock with respect to the satel- to minimize the estimation error, accurate knowledge of lite system time. From the principle of the constancy of the and t is needed. The former parameter is nowadays broad- speed of light c we have [1]: cast to final users with an accuracy between few dm and a meter thanks to accurate orbit determination techniques, a topic not addressed in this work. The estimation accuracy * Manuele Dassié of the latter parameter depends on the stability of the satel- manuele.dassie@dlr.de lite clocks and on their synchronization with respect to a Gabriele Giorgi common time scale (system time). In current GNSS pro- gabriele.giorgi@dlr.de cessing schemes is tightly coupled with the orbit determi- nation problem. Currently, inter-satellite clock offsets and Institute of Communications and Navigation, DLR, orbits are retrieved in a complex joint estimation process Wessling, Germany Vol.:(0123456789) 1 3 278 M. Dassié, G. Giorgi that requires a large ground network. The accuracy of the bodies. An analysis is carried out to obtain a model precise service provided at space segment level is usually charac- enough to satisfy the requirement of picosecond synchroni- terized in terms of Signal-in-Space Range Error (SiSRE), zation. For future use, this analysis also provides models for a parameter that factors in both orbit determination offsets femtosecond-level synchronization and beyond. and satellite clock errors and indicates the expected satellite- Finally, the third goal is to define the requirements on to-user range error. This error characterizes the maximum satellite orbit accuracy for correct relativistic modelling with ranging accuracy achievable by the system, which is then target performance of picosecond accuracy in synchroniza- subsequently further impaired by propagation (atmospheric) tion. This is done by analysing the magnitude of relativistic and user local effects (e.g. multipath and receiver biases). model errors induced by satellite orbit errors (sensitivity Typical SiSRE values in modern GNSSs are in the order analysis). of a few tens of centimeters. The DLR Institute of Com- munication and Navigation is currently working on a next generation satellite system (with codename Kepler) based on 2 Simultaneity optical inter-satellite links (OISLs) providing autonomous synchronization with offsets below 1 ps [5 , 6]. OISLs allow The notion of synchronization is closely connected with for a better separation of space (orbits) and time (synchro- the notion of simultaneity. Indeed, synchronized clocks nization), leading to a much higher level of synchronization must simultaneously produce the same time markers. The across the whole constellation than what is achievable in Newtonian theory of gravitation postulates the existence of current architectures, which in turn largely enhances precise absolute space and time, independent from each other. In the orbit determination. The SiSRE can potentially be reduced theory of relativity the refusal of the notions of absolute and to below one cm [9, 10], offering a highly accurate naviga- independent space and time results in different time rates in tion service to final users. different Reference Systems (RSs). As a consequence the The key to autonomous synchronization in space is the notion of simultaneity loses its absolute and unique mean- capability of remotely comparing satellite clock readings via ing [7]. OISLs. Two-Way Time Transfer techniques can be applied To address a problem in the framework of relativity it to retrieve relative clock offsets, but a careful modelling of is first necessary to introduce some concepts, such as the spurious effects introducing estimation biases is necessary. concepts of proper and coordinate quantities [16]: To the targeted level of inter-satellite synchronization of 1 ps, relativistic biases play a major role. These effects include Proper quantities are the direct results of observation second order Doppler shifts of clocks due to their veloc- without involving any information that is dependent on ity, gravitational shifts, and other relativistic delays on the the choice of a spacetime reference frame. In our analysis propagation of light through spacetime. If such effects are the most fundamental quantity is proper time, which is not properly accounted for, the benefit of operating stable the time measured by an observer in a frame of refer- clocks and having a mean for accurate synchronization via ence that is attached to the observer itself (proper to that OISLs will be masked [1]. Synchronization schemes for observer). picosecond-level accuracy need to be addressed in a rela- • Coordinate quantities are dependent on choices of a tivistic framework and this is the goal of this work. spacetime coordinate system. An example is the coordi- The first objective in this paper is to present a synchroni - nate time difference between two events (the difference zation method based on time transfer techniques. Satellites between the time coordinates of these events) or the rate will exchange electromagnetic signals containing informa- of a clock with respect to the coordinate time of some tion about times of emission and reception of reference bits spacetime reference system, which are both dependent which are taken in their proper time scale. The method pre- on the chosen reference system. sented will allow us to synchronize satellites pair-wise with respect to a defined coordinate time. The whole system can For analysis of any process in the framework of relativity be synchronized via consecutive synchronizations between one must introduce four-dimensional RSs. Consider a time satellite couples. scale  (denoting the proper time of an observer) and three The second objective is the characterization of the mag- space coordinates  =(x, y, z) . In the relativistic context nitude of the relativistic effects in order for the time transfer there are different proper time rates in different reference scheme to reach picosecond-level accuracy. Since this level systems. The relativistic synchronization framework in this of accuracy is rather high, higher-order relativistic effects work follows the concepts presented in [7]. Consider the need to be considered. This means studying the effect of following events: inclusion of Earth’s geopotential harmonics beyond the J moment, and analysing the influence of additional celestial Event E1 in RS1: ( ,  ) 1 1 1 3 Relativistic Modelling for Accurate Time Transfer via Optical Inter-Satellite Links 279 • Event E2 in RS2: ( ,  ) 2 2 In most cases, two different RSs refer to two different time scales  ≠  . This implies that we cannot say anything 1 2 about the simultaneity of the events E1 and E2. In order to determine if the two events are simultaneous we need to use the concept of coordinate time. Consider a third reference frame RS3 with proper time t and position coordinates  ̃ . For the other reference systems RS1 and RS2, this third reference frame can be used to define simultaneity. The events E1 and E2 can be expressed in coordinates Fig. 1 Coordinate time offset. Both clocks transform their time events of this third common RS as: to the coordinate scale. Given two arbitrary initial conditions the two clocks transform the time of the event to two different scales, one • Event E1 in RS3: (t ,  ̃ ) 1 1 shifted by  with respect to the other. We can see that the same AB • ̃ Event E2 in RS3: (t ,  ) observed event “Ev” is not simultaneous in the coordinate time scale. 2 2 Synchronizing the clocks means choosing well one of the initial con- ditions such that  = 0 or such that  is accurately known and AB AB We say E1 and E2 are simultaneous if the condition can be removed in post-processing t = t is satisfied in this common reference frame. This 1 2 0 0 third reference frame can be chosen freely, its choice depends on convenience and any RS can be used to define the coordinate instant t to the proper time instant  , we 0 0 can then unambiguously solve the differential Eq. (2 ) for coordinate time. Luckily, in most cases we can find a trans- formation that relates the proper time of an RS to the coor- the proper time instant (t) : dinate time. The coordinate scale t can be expressed as a function of the proper time scale  of the RS of interest, (t)=  + f (t, (t), (t))dt (3) i.e., t = t(). Using this notion, we can say that E1 and E2 are simul- 0 taneous if the condition t( )= t( ) is satisfied in the RS 1 2 0 0 Equation (3) can also be reversed: the coordinate time t where coordinate time is defined. This concept is what we can be expressed as a function of the proper time  as define as coordinate simultaneity. t = t() fixing a value for  . To have a solution we need to The key point is to find the relation linking the time scales ascribe a coordinate instant t to some proper instant  and 0 0 and  to the coordinate time scale t. If this relation is 1 2 solve Eq. (3) for t(). found it is possible to syntonize or, with an additional step, In the present work, we consider two coordinate time synchronize the clocks of two different reference systems scales t and t to be different even if they differ only by a with respect to the coordinate time. These two concepts need constant term: t = t + const . Consider two satellites A and to be defined [20]: B with proper times  and  . A B Coordinate syntonization: align the rate of clocks such • Setting an initial condition  (t )=  and using Eq. A 0 A that both run at same frequency. In the relativistic con- (3) for the specific case of A, it is possible to transform text to syntonize means to find the rate of a clock in an the readings of clock A to some coordinate time t. RS with respect to another clock in a different RS. Syn- In an analogous way it is possible to transform the read- tonizing a clock with proper time  with respect to the ings of clock B to some coordinate time t using Eq. (3) coordinate time t means finding the relation: for B and setting an initial condition  (t )=  . B B 0 0 = f (t, (t), (t)) (2) The coordinate time scale t differs from t due to the fact dt that the initial conditions are arbitrary and generally dif- where f is a function that depends on the coordinate time ferent. The two coordinate time scales t and t are related t, the position (t) and the velocity (t) of the clock. via t = t +  where  is a constant term that describes AB AB • Coordinate synchronization: this entails establishing the coordinate time offset between the two clocks. a one-to-one transformation between coordinate time To synchronize clocks A and B is to find a suitable ini- events t and proper time events  . By knowing the func- tial value  (t )=  for clock B enabling us to transform B 0 B tion f and assigning an initial condition (t )=  relating 0 0 the readings of B to the same coordinate time scale t as the 1 3 280 M. Dassié, G. Giorgi The Schwarzschild metric in isotropic coordinates one that A transforms to, i.e.  = 0 and t = t . A visual AB representation of these concepts is presented in Fig. 1. ( r, ,  ) is [11]: The initial conditions that allow for synchronization can GM ⎛ ⎞ be found with time transfer methods. The event that allows 1 − 2rc 2 2 2 ⎜ ⎟ −ds =− c dt us to ascribe t to  is called “the synchronizing event”. 0 B GM 0 ⎜ E ⎟ 1 + ⎝ ⎠ 2rc (4) � � � � GM 2 2 2 2 2 2 2.1 Choice of Coordinate Time + 1 + dr + r d + r sin d 2rc In general relativity every frame is equivalent so any RS −2 Expanding the parentheses and keeping only c terms we can be chosen to be the coordinate RS. Almost all users of get: GNSS are at fixed locations on the rotating Earth, or else are moving very slowly over Earth’s surface and the clocks are V v 2 2 2 ds ≈ 1 + − c dt (5) ticking according to a terrestrial time scale, typically UTC. 2 2 c 2c This leads to the natural idea of using an Earth-centered, GM Earth-fixed, reference frame (ECEF frame) as a coordinate where V(r)= − is the Earth’s monopole potential and RS. In this model the Earth rotates about a fixed axis with a 2 2 2 2 2 2 dr +r d +r sin d v ∶= is the square of the velocity of the −5 rotation rate of about  = 7.3 × 10 rad/s. Typically this dt clock. reference frame is the World Geodetic System (WGS-84) The spacetime metric is related to the proper time  of a [1]. The fact that this frame is rotating implies directly that clock via: the frame is non-inertial. However if the purpose is coor- dinate synchronization of multiple clocks the choice of the ds = cd (6) coordinate frame must be addressed carefully. In fact when With this, we can obtain the proper time rate of a clock with a non-inertial RS is chosen to define coordinate time there respect to ECI coordinate time as: is an absence of transitivity: if clock A is synchronized with clock B, and B with C, then A is not necessarily synchro- d V v = 1 + − . (7) nized with C. This gives rise to fundamental problems in the 2 2 dt c 2c synchronization of a GNSS. In order to overcome this diffi- culty one must construct in the neighbourhood of the Earth a The proper time rate in Eq. (7) is expressed with respect to local inertial RS, in which the gravitational field of external coordinate time t = t . ECI bodies can be represented in the form of tidal terms only The rate of coordinate time used in GNSS, however, is [7]. For GNSS it means that synchronization of the entire closely related to International Atomic Time (TAI), defined system of ground-based and orbiting clocks is performed in as the coordinate time at a surface of a constant effective the local inertial frame, or Earth Center Inertial (ECI) coor- gravitational equipotential at mean sea level in the ECEF. dinate system [2]. The coordinate time t of the ECI frame It is advantageous to exploit the fact that all clocks on the is defined at infinity, outside Earth’s gravity well [1 ]. If we reference surface tick at the same rate to redefine the rate of synchronize with respect to t = t , we are synchronizing ECI coordinate time by standard clocks at rest on Earth’s geoid. with respect to the coordinate time of an inertial RS, so that This is done by adopting the following coordinate change: the validity of transitivity can be assumed. dt = 1 + dt (8) Geoid ECI 3 The Proper Time Rate where t is the new coordinate time for clocks at rest on Geoid Earth’s geoid and  is the effective gravitational potential The relationship between the proper time  of a clock in the at the surface of the geoid (that also includes the Doppler vicinity of Earth and some coordinate time t is defined by term due to its rotation ) [2]. When this time scale change is the spacetime metric, which is derived from the metric ten- made, the proper time rate of Eq. (7) becomes: sor, the solution of the Einstein Field Equation. The metric V − d 0 v tensor completely characterizes the spacetime metric which = 1 + − . (9) 2 2 dt c 2c describes the relation between spatial and time coordinates. Geoid When Earth is modelled as a non-rotating spherical mass, We should remark here that in Eq. (9) we are comparing the the solution of such an equation leads to the Schwarzschild rate of a clock in the vicinity of Earth with respect to the metric. The Schwarzschild metric describes the region sur- time of a clock at rest on the surface of the geoid but with rounding a non-rotating spherical mass [17, 19]. 1 3 Relativistic Modelling for Accurate Time Transfer via Optical Inter-Satellite Links 281 Table 1 Values of used parameters [15] Parameter Notation Value −5 −1 Earth angular velocity 7.2921159 ×10 rad s −11 3 −1 −2 Universal gravitational G 6.67428 × 10 m kg s constant −1 Speed of light c 299792458 m s Earth mass M 5.972 × 10 kg WGS84 geoid semi-major a 6378136.6 m axis 2 −2 WGS84 effective potential  62636856.0 m s the understanding that synchronization is established in an underlying, locally inertial, reference frame (the ECI frame) [2]. The numerical value of the parameters and quantities presented in the preceding sections are given in Table 1. Fig. 2 Timing events: depicted are the trajectories of Satellite A and Satellite B with corresponding proper and coordinate times at signal transmission and reception in both directions of the communication 4 Autonomous Synchronization Some natural event, as an observation of occultation of a the following equation for the corresponding coordinate time star by a planet for instance, may be considered as a syn- t( ): chronizing event. Both observers may chose an arbitrary t( ) � � 2 initial condition linking their proper time to the coordinate V( (t )) −  v (t ) A 0 A −  = 1 + − dt (10) time, transform their proper time readings of observation of A A 2 2 c 2c the natural event into coordinate times t and t and then Ev Ev determine the coordinate offset  . A faster and more reli- AB Satellite A is thus able to transform its proper time of recep- able manner of synchronization is the artificial creation of tion  into the coordinate time instant t . If the trajectories A 3 a synchronizing event on the world line of the observer B of A and B are known with high accuracy, it is possible to via exchange of signals. This method of synchronization is determine the parameters  (t), v (t) and  (t), v (t) in the A A B B called “autonomous” or “independent” synchronization. In form of functions of the coordinate time scale t. Our goal is the system that we are considering, optical communication to retrieve the coordinate time of reception t so that we are enables the exchange of data with timestamps. This allows able to ascribe it to the reading  . The coordinate instant for time transfer between remote references [7]. t can be retrieved with the following iterative method [7]: 4.1 TwoW‑ay Synchronization (0) (0) t = t + t − t −  +  + (11) 0 3 0 Int hw 1 Prop Let’s consider a Two-Way Time Transfer (TWTT) scheme as presented in [7]: satellite A, located at  , transmits at A 1 (n) (n−1) t = t + t − t −  +  + (12) 0 3 0 Int hw 1 Prop coordinate time t a message with proper time stamp  . 0 A Satellite B receives it at coordinate time t , stamps it with proper time stamp  . After a delay satellite B re-transmits (n) t = lim t , n = 1, 2, 3, … 1 (13) it back stamping it with the proper time of re-transmission n→∞ . The signal is finally received at coordinate time t from B 3 where: Satellite A that stamps it at  . The coordinate propagation times for the round trip are respectively T = t − t and 2 AB 1 0 V( ()) −  v () B 0 B T = t − t . Assume that we have additional delays  and BA 3 2  = 1 − + d (14) Int 2 2 c 2c which denote the hardware delays relative to the reception (R) and transmission (T), with N = A, B . Figure 2 shows a model of this two-way exchange. Suppose we have an initial condition  (t )=  . Clock A A 0 A can transform its reading of any event happening at  solving 1 3 282 M. Dassié, G. Giorgi (n) (n) (n) 1 (n) (n) R R = T − T (15) (n) AB 0 AB Prop AB BA T = + AB 2 c c c � � (n) (23) ‖ (t )‖ + ‖ (t )‖ + R 2GM A 0 B AB (n) (n) E 1 R R + ln (n) AB (n) BA 3 (n) (16) T = , T = ‖ (t )‖ + ‖ (t )‖ − R AB BA A 0 B AB c c (0) Here, T is the roughly estimated propagation time that can AB (n) (n) R = ‖ (t )−  (t )‖ be computed thanks to the approximate knowledge of the (17) B A 0 AB 1 positions of the satellites. The first term on the right-hand side of Eq. (23) is the (n) (n) R = ‖ (t )−  (t +  )‖ (18) A 3 B Int BA 1 classical Euclidean time that light takes to go from point A to point B. The second and third terms are delays resulting 1 1 T R R T from the time dilation experienced by light when travel- =  −  +  − (19) hw A A B B 2 2 ling through a gravitational field and are purely relativistic effects. The third term is called the Shapiro delay. This Here,  is the integral of the proper time rate over the path Int expression for the propagation of light can be found by of clock B between reception from A and transmission from setting the spacetime element ds = 0 , solving for dt and B, which in coordinate time corresponds to t − t . The term 2 1 integrating it on the path from A to B. The two relativistic is the half difference of propagation time during the Prop terms are of the order of tens of picoseconds and therefore round trip between A and B (expressed with T and T AB BA (0) have to be taken into account to achieve picosecond-level respectively). The initial  can be estimated thanks to the Prop synchronization with an OWTT. Note also that reception approximate knowledge of the positions of the satellites. event t is determined with the knowledge of the trajecto- Finally,  identifies the residual hardware delay in a TWTT hw ries  (t),  (t) . In an OWTT, inaccurate knowledge of the A B exchange. This may be largely mitigated if the transmit and positions and trajectories would directly result in an offset receive delays at either side of the communication are simi- due to errors in the calculation of T . This is one of the AB lar. Satellite B observes the signal reception happening at main limitations of this method: before synchronization, . Thus, we can ascribe the calculated moment t to the B 1 even assuming that the satellite positions are known with observed moment  and, thereby synchronize clock B with cm accuracy, the resulting modelling errors are in the order clock A. We found the initial condition  (t )=  that B 1 B of hundreds of picoseconds in the determination of t (and allows clock B to transform all events seen in its timescale thus in a corresponding synchronization offset). Moreover, to the same coordinate time t as the one that clock A in an OWTT we can only achieve clock synchronization transforms to. if spurious terms, such as hardware delays in the optical terminals used to establish the link, can be modelled and removed. This may prove difficult: a characterisation of 4.2 Comparison with a OneW ‑ ay Synchronization these delays in a relevant operational environment would be difficult to achieve for picosecond-level accuracy. For completeness, we could consider a One-Way Time Due to the quasi-symmetry of the communication, a Transfer (OWTT), where we have a unidirectional TWTT is preferred because it mitigates all the problems exchange of signals from Satellite A to Satellite B. In of the OWTT mentioned above. Positioning errors, hard- Fig. 2 the communication would end at the signal recep- ware delays and additional relativistic offsets mitigate or tion from Satellite B at coordinate time instant t . In this cancel out in the expressions for determining t . I n de e d , case t can be determined with the following iterative in a TWTT we are interested in characterizing the differ - method: ence of propagation times during the round trip and not (0) (0) their absolute magnitude. If we compute Eq. (15) using the t = t + T (20) 1 AB expression for the one-way propagation times (23) we can see that the difference of the additional relativistic terms (n) (n−1) t = t + T (21) is of the order of femtoseconds and is therefore negligible. 1 AB In fact in Eq. (16) the expression for the propagation time (n) is simpler and only includes the Euclidean term. It will be t = lim t , n = 1, 2, 3, … (22) n→∞ shown in a dedicated section (Sect. 6.2.2) that positioning errors are also mitigated in a TWTT. where: 1 3 Relativistic Modelling for Accurate Time Transfer via Optical Inter-Satellite Links 283 n n max 4.3 System Synchronization GM V (r, 𝛼 , 𝛽 ) =− 1 + P (sin 𝛽 ) E nm r r n=2 m=0 One can synchronize any arbitrary number of other satellites (25) using a cascaded approach, i.e. Satellite A gets synchronized ̄ ̄ × C cos(m𝛼 )+ S sin(m𝛼 ) nm nm with B via a time transfer, B gets synchronized with C and so on until all satellites are synchronized to the same coor- where: dinate scale. However, this implies that at least one clock exchanges signals with a clock keeping the coordinate time. • a is the semi-major axis of the WGS 84 Ellipsoid; In a GNSS this last clock would be on Earth. Essentially, ̄ ̄ • C and S are the normalized gravitational coefficients nm nm only one ground station would be sufficient to guarantee the contained in EGM2008; link with the terrestrial time scale. The presented synchro- • P is the normalized associated Legendre function of nm nization method allows for a significant degree of auton- degree n and order m; omy of the satellite system. In the extreme case, lacking and  are respectively the longitude and latitude of the even the ground station, the satellite clocks realize a “space satellite position in an ECEF frame. clock” whose time scale definition is not affected by the poor knowledge of the geopotential on the surface of Earth Not all terms in the summation are significant: the model [20]. Such a clock, moving in an unperturbed orbit and well- should only retain n terms, sufficient to guarantee that the max defined coordinate system, would bring significant value and model inaccuracies remain below ps-level. The term V is SC new capabilities to time and frequency transfer worldwide, the gravitational contribution of body C (mass M ) situated to precision geodesy and terrestrial reference frames, to at  on satellite S situated at  . This is expressed as [20]: earth-environmental science and to navigation systems [3]. 1 1 V =−GM − − (26) SC C r r SC C 5 Picosecond Synchronization Models C where  =  −  and r = ‖‖. In both time transfer methods presented, in order to reach The relation between the rate of a clock in the near Earth picosecond precision it is necessary to have precise models region (proper time  ) with respect to coordinate time on of the proper time rate and of the propagation time T AB dt Earth’s geoid becomes: (or the difference of propagation times in a TWTT). V  V d v E 0 SC ≈ 1 + − − + (27) 2 2 2 2 dt c c 2c c 5.1 T he proper Time Integral C≠E We recall here the expression of the proper time rate of a 5.2 Evaluation of Proper Time Rate Terms for Kepler clock in Earth’s vicinity with respect to TAI coordinate time Satellites t: V((t)) − d(t) v(t) In this section, we are interested in determining the number = 1 + − (24) 2 2 dt c 2c of terms to be included in the proper time rate expression to be integrated in order for the integral terms to meet dt Integrating Eq. (24) it is possible to transform any reading of the requirement of picosecond-level accuracy. We need to the clock  to events of coordinate time scale t. This integral characterize which effects play a role within the time of a has to be solved analytically or numerically in order to relate signal exchange, which in the case of typical GNSS satellites instants of the two scales. The complexity of finding a solu- in Medium Earth Orbit (MEO) is always within a second. tion depends on the model of the potential. The gravitational The Kepler constellation consists of two segments: a set of potential V in Eq. (24) is the sum of the Earth’s gravitational navigation satellites in MEO, assigned approximately to the potential V and the one created by the cumulative contribute same orbital slots of the current Galileo constellation, and a of other celestial planets, V . SC smaller set of satellites in upper Low Earth Orbit (LEO). The The Earth’s gravitational potential V is defined through relevant orbital parameters are summarized in Table 2. a summation of spherical harmonics [14]: A list of coefficients, degrees and orders, with up to 2190 degrees, can be found at [13]. 1 3 284 M. Dassié, G. Giorgi Table 2 Summary of main Kepler parameters Table 3 Number of geopotential harmonic terms to take into account in the proper time rate for different thresholds Segment Type Inclination Semi-major axis Constellation Threshold [s/s] n max MEO Walker 24/3/1 29601.3 km −12 ◦ LEO 2 (3 LEO Walker 6/2/1 7626.3 km 89.7219 recom- mended) −15 −18 −12 MEO 2 −15 −18 10 5 We can see in Fig.  3 that only two terms (quadrupole formulation of the potential) are sufficient to describe the proper time rate of satellites in both constellations at the −12 order of 10 . From the third term the information apported can be considered negligible. In any case it is clearly visible that the geopotential term of order n = 2 for LEO satellites is very close to this threshold. In order to keep some margin it is advisable to consider a potential expansion with n = 3 max for the LEO constellation. As expected more terms in the Fig. 3 Maximum contribution of Earth’s potential harmonic terms of potential expansion are needed for stricter thresholds. A degree n on MEO and LEO satellites proper time rates summary of the results is presented in Table 3. Using the orbital simulation data from GMAT each term The satellites of both segments are homogeneously dis- of Eq. (27) for every single satellite has been computed and the maximal contribution of each term has been recorded. tributed over their orbital planes, i.e. 45 difference in mean anomaly for MEO and 60 for LEO neighbour satellites. The orbital parameters of the sun, moon and other planets have been artificially modified to simulate the case where The planes are also homogeneously distributed around the globe with a 120 separation in the right ascension of the every other body considered is at its closest approach during the whole simulation. This is an extreme case that is very ascending node (RAAN) for MEO planes and 90 for LEO planes. The argument of perigee is 0 for both segments. unlikely to happen but it gives an upper bound to the shift caused by each one of those bodies. Since the communication time is always shorter than 1 s we can directly look for terms in the proper time rate that are Figure 4 shows the magnitude of the maximal contribu- −12 tions of the terms in Eq. (27) for MEO and LEO satellites smaller than the threshold of 10 s/s. For completeness and future use we evaluate all terms of the proper time rate in the constellation. For picosecond accuracy only the geo- −12 −15 potential term and the second order Doppler term need to be for different satellites against 3 thresholds: 10 , 10 and −18 10 s/s. The satellites’ orbits are simulated using the open considered. Gravitational effects of external bodies need to −15 be taken into account for the stricter thresholds of 10 s/s source software General Mission Analysis Tool (GMAT), −18 available from NASA [12]. The simulation interval is chosen (Moon influence on MEO satellites) and 10 s/s (Moon and Sun for both MEO and LEO satellites). Table 4 summarizes to be the 10 days and the integration step to propagate the orbits is 30 s. all the terms to be taken into account for a given constella- tion and threshold. We have presented in Eq. (27) the terms to consider for a full model of the proper time rate for a clock located at (t) in the vicinity of the Earth with respect to TAI coordi- nate time. As presented in Eq. (25), the “geopotential” V 6 Impact of Orbit Parameter Errors (gravitational potential of Earth) is defined through a sum- mation of spherical harmonics. We name each term of the We are now interested in the sensitivity of the derived mod- geopotential V . Figure 3 shows the maximal contribution of the single els to orbit parameter errors. To analyse it we evaluate the resulting errors in proper time rate and in the difference of geopotential terms of degree n manifested during the simulated 10 days. propagation times arising from errors in position, range and 1 3 Relativistic Modelling for Accurate Time Transfer via Optical Inter-Satellite Links 285 Table 5 Allowed velocity error Orbit H  [m/s] to remain below a proper time rate error threshold H −12 LEO 12.42 −15 −3 10 12.42 × 10 −18 −6 10 12.43 × 10 −12 MEO 10 24.40 −3 −15 24.48 × 10 −18 −6 10 24.48 × 10 (v + ) d d 0 0 (30) − = − = H 2 2 dt dt 2c 2c w/Error w/o Error which leads to: = v + 2Hc − v (31) Fig. 4 Maximum value of terms in the proper time rate expression for Assuming circular orbits, the orbital velocity of a body is LEO and MEO satellites during the simulated 10 days GM v = where a is the orbit radius. Table 5 summarizes the results for all thresholds H and both types of satellites Table 4 Terms to take into account in the proper time rate for differ - considered for the Kepler constellation. It is noticeable how ent thresholds the requirement for the uncertainty in the velocity is techno- Constellation Threshold [s/s] Terms −12 logically not hard to meet (at least for the 10 case). The −12 velocity can be determined with a higher level of precision LEO 10 Geopot, Doppler without major difficulties. −15 10 Geopot, Doppler −18 Geopot, Doppler, Moon, Sun −12 6.2 Position Error Margin MEO Geopot, Doppler −15 Geopot, Doppler, Moon 6.2.1 Impact of Position Errors on the Proper Time Rate −18 Geopot, Doppler, Moon, Sun Satellite position errors impact the determination of the proper time rate because of the dependence of the Earth’s velocity. In fact we want to determine the permitted error on geopotential on position. Particularly affected is the uni - those parameters that leads to: form geopotential term, which is the dominant term in the d d expansion and depends on the norm of the position vector − = H [s∕s] (28) (t) . Consider (r + ) the norm of the position vector of the dt dt w/Error w/o Error 0 satellite with error and r the same quantity without error: � � −  = H [s] � � V V Prop Prop �w/Error �w/o Error − = H (32) 2 2 c c w/Error w/o Error ⎧ −12 (29) ⎪ GM −15 Using the assumption of uniform gravity V(r)= − : where H = 10 . −18 ⎩ GM 1 1 − + = H (33) c r +  r o o which leads to: 6.1 Velocity Error Margin −1 Hc 1 = − + − r (34) An error in the velocity of the satellites would impact the GM r E 0 determination of the proper time rate, affecting the second- Table 6 summarizes the results for all thresholds H evaluat- order Doppler term. Consider (v + ) the velocity of the ing Eq. (34) for both types of satellites of the Kepler con- satellite with error and v the velocity without error: stellation. It can be seen again how the requirement for the 1 3 286 M. Dassié, G. Giorgi Table 6 Permitted radial Table 7 Clock offsets resulting from 4 m positioning errors in TWTT Orbit H  [m] position error to remain below a assuming that those errors remain constants during the whole signal proper time rate error threshold −12 3 exchange process LEO 10 13.136 × 10 −15 13.1149 Communication t − t  [s] |H| [s] 2 0 −18 −3 10 13.114 × 10 −18 MEO-MEO neighbours 0 9.8986 × 10 −12 3 MEO 10 198.900 × 10 −13 0.1 1.6545 × 10 −15 10 197.573 −13 0.5 8.2728 × 10 −18 −3 10 197.572 × 10 −17 MEO-MEO next-neighbours 0 1.2348 × 10 −13 0.1 1.6544 × 10 −13 0.5 8.2723 × 10 −12 uncertainty in the positioning is very loose (for the 10 s/s −13 LEO-MEO 0 3.4792 × 10 case). There is no difficulty in knowing the radial position −13 0.1 3.2474 × 10 with such precision. −13 0.5 3.4630 × 10 6.2.2 Impact of Position Errors on the Propagation Time The offset is minimal when  ≈− . This is the case for The real impact of satellite position errors manifests itself in AB BA an approximately simultaneous exchange of signals, where the calculation of the difference in propagation times, rather t ≈ t . In practice, since the clocks will be roughly synchro- than on the proper time rate. Assume that the real positions 0 2 nized, both Satellite A and Satellite B will send a signal at (t) and  (t) are affected by positioning errors (in vectorial A B proper times  and  which they can approximately relate form) expressed respectively as  (t) and  (t) due to uncer- A0 B2 A B to coordinate times t ≈ t . tainties in the orbit determination. The difference in propaga- 0 2 The worst case scenario is when  =− and both tion times is then: A B uncertainties are parallel to  +  . Preliminary analy- AB BA ‖ +  (t )−  (t )‖ � AB B 1 A 0 sis of Precise Orbit Determination (POD) for the Kepler Prop �w/Error 2c constellation shows that by considering a simultaneous (35) ‖ +  (t )−  (t )‖ BA A 3 B 2 application of a large number of modelling errors (with- 2c out using error reduction techniques), satellite 3D posi- tion errors are expected to be smaller than 70 cm. In the As a first approximation we can assume that the posi- mentioned case, POD is performed by a single ground tion errors remain basically constant during the com- station with the support of the LEO segment, without any munication period which is expected to last a few hun- assumption of prior synchronization or use of ISLs [9]. In dreds of ms, i.e.  ∶=  (t )≈  (t )≈ const. A A 0 A 3 an operational scenario, satellites would infer their posi- and  ∶=  (t )≈  (t )≈ const. We have that B B 1 B 2 tion from the broadcast ephemerides based on predicted ‖ −  ‖ ≪ ‖ ‖ so we can use the Taylor expansion B A AB orbits. The accuracy of the predicted orbits is expected to of Eq. (35). be lower than the one of POD orbits but it is nevertheless Thus: expected to be much better than 1 meter. To remain con- R R servative, we assume here that the positions of all com- AB BA ≈ − Prop municating Kepler satellites are known with a relatively w/Error c c (36) large uncertainty ‖ ‖ = ‖ ‖ = 4 m. A simulation has A B +   − AB BA B A been performed by injecting orbital errors at this level and 2c the results are given in Table 7. In the simulation different types of transmission delays t − t are evaluated. In the 2 0 =∶   + H first case optical signals are transmitted simultaneously, (37) Prop w/o Error i.e. t = t . In the other cases Satellite B transmits after 0 2 where H corresponds to the last term on the right hand side delay t − t with respect to the moment of transmission 2 0 AB BA of Eq. (36),  = and  = . t from Satellite A. We can see from the results that even AB BA 0 ‖ ‖ ‖ ‖ AB BA when a large uncertainty in the positions of the satellites We identify here a synchronization offset H due to uncer- is considered and the communication is very asymmetric tainty in orbital position determination, expressed as: ( t > t ), we can still model relativistic effects to within 1 2 0 +   −  ps accuracy. This means that with the current orbit deter- AB BA B A (38) H = mination capabilities we can accurately model relativistic 2c effects to below 1 ps. 1 3 Relativistic Modelling for Accurate Time Transfer via Optical Inter-Satellite Links 287 and 3 we have defined simultaneity and coordinate syn- 7 Technological State of the Art chronization in the relativistic domain. A synchronization method based on a Two-Way Time Transfer was presented So far, time transfer has been presented in abstract form in Sect.  4. Then the relativistic effects to be taken into through a mathematical description of the process. The goal account for a time transfer with the necessary degree of of this section is to provide an overview of the technology accuracy have been studied, and Sect.  5 shows that the required to make the presented synchronization methods a gravitational influence of other celestial bodies can be reality. Two-way time transfer at ps-level is possible via com- neglected and only the Earth geopotential has to be taken munication systems based on lasers, already space-qualified into account. The Earth’s geopotential needs to be char- and operative for commercial satellite data relay applications acterised with a higher degree of accuracy than in cur- [5]. Early ground-to-space tests of time transfer via non-coher- rent GNSS systems, considering harmonics beyond the J ent optical links have already demonstrated accuracies in the moment. For a correct relativistic modelling with the goal range of hundreds picoseconds. Examples of such tests are of picosecond-level synchronization accuracy, there is a the Laser Time Transfer technology validated at 300 ps-level need to keep the errors on orbits below a certain threshold. on Beidou satellites [8] and the Time Transfer by Laser Link These thresholds have been calculated and presented in (T2L2), which demonstrated time transfer between a number Sect. 6. It was discussed how a Two-Way Time Transfer of International Laser Ranging Service (ILRS) stations and the picosecond synchronization is not only fundamentally fea- Jason-2 satellite below the 100 ps-level [4]. These methods use sible, but can also be implemented in practice. non-coherent links. By exploiting laser-based coherent links, a A relativistic analysis is a fundamental step in the defi- network of two or more frequency references synchronized at nition of the GNSS processing architecture, since in this sub-ps level can be established. A DLR laboratory demonstra- framework synchronization offsets derive from fundamen- tor of coherent links is being developed to verify optical range, tal characteristics of nature. The feasibility of relativistic time transfer, and data transmission. The demonstrator consists synchronization with OISLs confirms the possibility of of two terminals, running a bidirectional free-space optical separating space and time in the error estimation process, link in the laboratory, with single-mode fiber coupling in the resulting in a much higher level of synchronization across receivers at both sites. The optical carrier is modulated by a the constellation than is achievable in current architectures. fast ranging sequence and a slower data stream. The coherent This has the potential to greatly improve accurate orbit transceivers enable time-stamping received reference bits with determination and consequently the navigation service for sub-ps precision, enabling the exchange of measured time- end users. Furthermore, the significant degree of autonomy of-arrival information between the paired satellites, which in synchronization enabled by the proposed TWTT scheme in turns enable two-way time transfer to retrieve clock off- has a potentially large impact on infrastructure and opera- sets and perform inter-satellite ranging at sub-mm level [18]. tional costs. The feasibility of an innovative concept of a Such optical transceivers could be integrated in existing opti- fully autonomous ’space clock’ was also addressed. The idea cal communication terminals, such as those employed in the of such a clock could be further investigated and has the SpaceDataHighway operated in space since a few years in the potential to strongly impact the the field of time metrology, framework of the Copernicus program to optically link LEO with the definition of a spatial time scale less affected by and GEO satellites. This integration is the focus of a DLR various terrestrial phenomena. mission named COMPASSO, initiated in 2021, aiming at test- ing a coherent optical link for time/frequency transfer, rang- Acknowledgements I would like to thank my advisor Dr. Gabriele ing and data exchange between a terminal on the international Giorgi for his excellent supervision both during the thesis and after- space station and an optical ground station in Europe. The wards. It is thanks to his support that this work has seen the light of day. mission will hopefully pave the way to further validation mis- In addition to my advisor, I would like to thank the director of the DLR Institute of Communications and Navigation, Christoph Günther, for sions aimed at testing autonomous accurate synchronization the support and nice discussion regarding this work. I would also like to between satellites in higher orbits, and future utilization in thank my parents and aunt, who have always supported me throughout GNSS constellations to augment navigation solution accuracy, my academic career and my life in general. system operability and robustness of operations. Funding Open Access funding enabled and organized by Projekt DEAL. This work was done in the context of the ADVANTAGE pro- ject, supported by the Helmholtz-Gemeinschaft Deutscher Forschun- 8 Conclusion and Future Perspectives gszentren e.V. (Helmholtz Association of German Research Centers) under grant number ZT-0007, and in the OTTEx project, supported This work focused on analysing how to accurately model by the European Space Agency (grant number H2020-ESA-038.04). relativistic effects to enable ps-accurate time transfer Availability of Data and Material Not applicable. between satellites in MEO and LEO orbits. In Sects. 2 1 3 288 M. Dassié, G. Giorgi Code availability Custom code. and experiment of onboard laser time transfer in chinese beidou navigation satellites. Adv. Space Res. 51(6), 951–958 (2013). https:// doi. org/ 10. 1016/j. asr. 2012. 08. 007 Declarations 9. Michalak, G., Glaser, S., Neumayer, K., König, R.: Precise orbit and earth parameter determination supported by leo satellites, Conflict of interest On behalf of all authors, the corresponding author inter-satellite links and synchronized clocks of a future gnss. Adv. states that there is no conflict of interest. Space Res. (2021) 10. 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Relativistic Modelling for Accurate Time Transfer via Optical Inter-Satellite Links

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Abstract

The DLR Institute for Communication and Navigation is currently working on a new GNSS architecture that enables accurate autonomous inter-satellite synchronization at picosecond-level. Synchronization is achieved via time transfer techniques enabled by optical inter-satellite links (OISLs), paving the way for a system in which space (orbits) and time (synchroniza- tion) can be effectively separated, leading to a high level of synchronization throughout the constellation, which in turn greatly improves accurate orbit determination. This is possible provided that relativistic effects are adequately taken into account. This work focuses on a two-way time transfer scheme based on the exchange of time stamps via optical signals, which allows the synchronization of a GNSS satellite system with respect to a defined coordinate time with picosecond-level accuracy. We analyse the impact of relativistic effects in clock offset estimation between optically linked clocks: results show that to achieve synchronization at this level of accuracy it is necessary to account for terrestrial geopotential harmonics up to the third order while the gravitational influence of additional celestial bodies can be neglected. Relativistic delays in the propagation of electromagnetic waves through spacetime are also evaluated. It is shown that for a two-way synchronization method, the Euclidean expression for the propagation of light is sufficient to achieve picosecond synchronization, provided m-level orbit determination of both satellites is available, and the hardware delays are well calibrated to the targeted accu- racy. Also, we show how to practically achieve autonomous synchronization via a sequence of pair-wise synchronizations across all satellites of the constellation. Keywords Time transfer · Proper time rate · Intersatellite links · Relativistic propagation 1 Introduction c t + t − t =  −  , j = 1, 2, 3, 4 (1) j j Position determination and time transfer in GNSS rely on a Having four independent equations, one can solve for the simple model. Consider four synchronized clocks located at four unknowns , t . Such a solution is also called a “navi- positions  , j = 1, 2, 3, 4 . The satellites transmit electromag- gation solution”. In principle the solution can be computed netic pulses at times t respectively. Suppose that these four with information from any number of satellites greater than signals are received at position  at instant t. The receiver or equal to four. Timing errors of 1 ns will lead to position- records the reception event happening at t + t where t is ing errors in the order of 30 cm. It is obvious that in order the offset of the receiver clock with respect to the satel- to minimize the estimation error, accurate knowledge of lite system time. From the principle of the constancy of the and t is needed. The former parameter is nowadays broad- speed of light c we have [1]: cast to final users with an accuracy between few dm and a meter thanks to accurate orbit determination techniques, a topic not addressed in this work. The estimation accuracy * Manuele Dassié of the latter parameter depends on the stability of the satel- manuele.dassie@dlr.de lite clocks and on their synchronization with respect to a Gabriele Giorgi common time scale (system time). In current GNSS pro- gabriele.giorgi@dlr.de cessing schemes is tightly coupled with the orbit determi- nation problem. Currently, inter-satellite clock offsets and Institute of Communications and Navigation, DLR, orbits are retrieved in a complex joint estimation process Wessling, Germany Vol.:(0123456789) 1 3 278 M. Dassié, G. Giorgi that requires a large ground network. The accuracy of the bodies. An analysis is carried out to obtain a model precise service provided at space segment level is usually charac- enough to satisfy the requirement of picosecond synchroni- terized in terms of Signal-in-Space Range Error (SiSRE), zation. For future use, this analysis also provides models for a parameter that factors in both orbit determination offsets femtosecond-level synchronization and beyond. and satellite clock errors and indicates the expected satellite- Finally, the third goal is to define the requirements on to-user range error. This error characterizes the maximum satellite orbit accuracy for correct relativistic modelling with ranging accuracy achievable by the system, which is then target performance of picosecond accuracy in synchroniza- subsequently further impaired by propagation (atmospheric) tion. This is done by analysing the magnitude of relativistic and user local effects (e.g. multipath and receiver biases). model errors induced by satellite orbit errors (sensitivity Typical SiSRE values in modern GNSSs are in the order analysis). of a few tens of centimeters. The DLR Institute of Com- munication and Navigation is currently working on a next generation satellite system (with codename Kepler) based on 2 Simultaneity optical inter-satellite links (OISLs) providing autonomous synchronization with offsets below 1 ps [5 , 6]. OISLs allow The notion of synchronization is closely connected with for a better separation of space (orbits) and time (synchro- the notion of simultaneity. Indeed, synchronized clocks nization), leading to a much higher level of synchronization must simultaneously produce the same time markers. The across the whole constellation than what is achievable in Newtonian theory of gravitation postulates the existence of current architectures, which in turn largely enhances precise absolute space and time, independent from each other. In the orbit determination. The SiSRE can potentially be reduced theory of relativity the refusal of the notions of absolute and to below one cm [9, 10], offering a highly accurate naviga- independent space and time results in different time rates in tion service to final users. different Reference Systems (RSs). As a consequence the The key to autonomous synchronization in space is the notion of simultaneity loses its absolute and unique mean- capability of remotely comparing satellite clock readings via ing [7]. OISLs. Two-Way Time Transfer techniques can be applied To address a problem in the framework of relativity it to retrieve relative clock offsets, but a careful modelling of is first necessary to introduce some concepts, such as the spurious effects introducing estimation biases is necessary. concepts of proper and coordinate quantities [16]: To the targeted level of inter-satellite synchronization of 1 ps, relativistic biases play a major role. These effects include Proper quantities are the direct results of observation second order Doppler shifts of clocks due to their veloc- without involving any information that is dependent on ity, gravitational shifts, and other relativistic delays on the the choice of a spacetime reference frame. In our analysis propagation of light through spacetime. If such effects are the most fundamental quantity is proper time, which is not properly accounted for, the benefit of operating stable the time measured by an observer in a frame of refer- clocks and having a mean for accurate synchronization via ence that is attached to the observer itself (proper to that OISLs will be masked [1]. Synchronization schemes for observer). picosecond-level accuracy need to be addressed in a rela- • Coordinate quantities are dependent on choices of a tivistic framework and this is the goal of this work. spacetime coordinate system. An example is the coordi- The first objective in this paper is to present a synchroni - nate time difference between two events (the difference zation method based on time transfer techniques. Satellites between the time coordinates of these events) or the rate will exchange electromagnetic signals containing informa- of a clock with respect to the coordinate time of some tion about times of emission and reception of reference bits spacetime reference system, which are both dependent which are taken in their proper time scale. The method pre- on the chosen reference system. sented will allow us to synchronize satellites pair-wise with respect to a defined coordinate time. The whole system can For analysis of any process in the framework of relativity be synchronized via consecutive synchronizations between one must introduce four-dimensional RSs. Consider a time satellite couples. scale  (denoting the proper time of an observer) and three The second objective is the characterization of the mag- space coordinates  =(x, y, z) . In the relativistic context nitude of the relativistic effects in order for the time transfer there are different proper time rates in different reference scheme to reach picosecond-level accuracy. Since this level systems. The relativistic synchronization framework in this of accuracy is rather high, higher-order relativistic effects work follows the concepts presented in [7]. Consider the need to be considered. This means studying the effect of following events: inclusion of Earth’s geopotential harmonics beyond the J moment, and analysing the influence of additional celestial Event E1 in RS1: ( ,  ) 1 1 1 3 Relativistic Modelling for Accurate Time Transfer via Optical Inter-Satellite Links 279 • Event E2 in RS2: ( ,  ) 2 2 In most cases, two different RSs refer to two different time scales  ≠  . This implies that we cannot say anything 1 2 about the simultaneity of the events E1 and E2. In order to determine if the two events are simultaneous we need to use the concept of coordinate time. Consider a third reference frame RS3 with proper time t and position coordinates  ̃ . For the other reference systems RS1 and RS2, this third reference frame can be used to define simultaneity. The events E1 and E2 can be expressed in coordinates Fig. 1 Coordinate time offset. Both clocks transform their time events of this third common RS as: to the coordinate scale. Given two arbitrary initial conditions the two clocks transform the time of the event to two different scales, one • Event E1 in RS3: (t ,  ̃ ) 1 1 shifted by  with respect to the other. We can see that the same AB • ̃ Event E2 in RS3: (t ,  ) observed event “Ev” is not simultaneous in the coordinate time scale. 2 2 Synchronizing the clocks means choosing well one of the initial con- ditions such that  = 0 or such that  is accurately known and AB AB We say E1 and E2 are simultaneous if the condition can be removed in post-processing t = t is satisfied in this common reference frame. This 1 2 0 0 third reference frame can be chosen freely, its choice depends on convenience and any RS can be used to define the coordinate instant t to the proper time instant  , we 0 0 can then unambiguously solve the differential Eq. (2 ) for coordinate time. Luckily, in most cases we can find a trans- formation that relates the proper time of an RS to the coor- the proper time instant (t) : dinate time. The coordinate scale t can be expressed as a function of the proper time scale  of the RS of interest, (t)=  + f (t, (t), (t))dt (3) i.e., t = t(). Using this notion, we can say that E1 and E2 are simul- 0 taneous if the condition t( )= t( ) is satisfied in the RS 1 2 0 0 Equation (3) can also be reversed: the coordinate time t where coordinate time is defined. This concept is what we can be expressed as a function of the proper time  as define as coordinate simultaneity. t = t() fixing a value for  . To have a solution we need to The key point is to find the relation linking the time scales ascribe a coordinate instant t to some proper instant  and 0 0 and  to the coordinate time scale t. If this relation is 1 2 solve Eq. (3) for t(). found it is possible to syntonize or, with an additional step, In the present work, we consider two coordinate time synchronize the clocks of two different reference systems scales t and t to be different even if they differ only by a with respect to the coordinate time. These two concepts need constant term: t = t + const . Consider two satellites A and to be defined [20]: B with proper times  and  . A B Coordinate syntonization: align the rate of clocks such • Setting an initial condition  (t )=  and using Eq. A 0 A that both run at same frequency. In the relativistic con- (3) for the specific case of A, it is possible to transform text to syntonize means to find the rate of a clock in an the readings of clock A to some coordinate time t. RS with respect to another clock in a different RS. Syn- In an analogous way it is possible to transform the read- tonizing a clock with proper time  with respect to the ings of clock B to some coordinate time t using Eq. (3) coordinate time t means finding the relation: for B and setting an initial condition  (t )=  . B B 0 0 = f (t, (t), (t)) (2) The coordinate time scale t differs from t due to the fact dt that the initial conditions are arbitrary and generally dif- where f is a function that depends on the coordinate time ferent. The two coordinate time scales t and t are related t, the position (t) and the velocity (t) of the clock. via t = t +  where  is a constant term that describes AB AB • Coordinate synchronization: this entails establishing the coordinate time offset between the two clocks. a one-to-one transformation between coordinate time To synchronize clocks A and B is to find a suitable ini- events t and proper time events  . By knowing the func- tial value  (t )=  for clock B enabling us to transform B 0 B tion f and assigning an initial condition (t )=  relating 0 0 the readings of B to the same coordinate time scale t as the 1 3 280 M. Dassié, G. Giorgi The Schwarzschild metric in isotropic coordinates one that A transforms to, i.e.  = 0 and t = t . A visual AB representation of these concepts is presented in Fig. 1. ( r, ,  ) is [11]: The initial conditions that allow for synchronization can GM ⎛ ⎞ be found with time transfer methods. The event that allows 1 − 2rc 2 2 2 ⎜ ⎟ −ds =− c dt us to ascribe t to  is called “the synchronizing event”. 0 B GM 0 ⎜ E ⎟ 1 + ⎝ ⎠ 2rc (4) � � � � GM 2 2 2 2 2 2 2.1 Choice of Coordinate Time + 1 + dr + r d + r sin d 2rc In general relativity every frame is equivalent so any RS −2 Expanding the parentheses and keeping only c terms we can be chosen to be the coordinate RS. Almost all users of get: GNSS are at fixed locations on the rotating Earth, or else are moving very slowly over Earth’s surface and the clocks are V v 2 2 2 ds ≈ 1 + − c dt (5) ticking according to a terrestrial time scale, typically UTC. 2 2 c 2c This leads to the natural idea of using an Earth-centered, GM Earth-fixed, reference frame (ECEF frame) as a coordinate where V(r)= − is the Earth’s monopole potential and RS. In this model the Earth rotates about a fixed axis with a 2 2 2 2 2 2 dr +r d +r sin d v ∶= is the square of the velocity of the −5 rotation rate of about  = 7.3 × 10 rad/s. Typically this dt clock. reference frame is the World Geodetic System (WGS-84) The spacetime metric is related to the proper time  of a [1]. The fact that this frame is rotating implies directly that clock via: the frame is non-inertial. However if the purpose is coor- dinate synchronization of multiple clocks the choice of the ds = cd (6) coordinate frame must be addressed carefully. In fact when With this, we can obtain the proper time rate of a clock with a non-inertial RS is chosen to define coordinate time there respect to ECI coordinate time as: is an absence of transitivity: if clock A is synchronized with clock B, and B with C, then A is not necessarily synchro- d V v = 1 + − . (7) nized with C. This gives rise to fundamental problems in the 2 2 dt c 2c synchronization of a GNSS. In order to overcome this diffi- culty one must construct in the neighbourhood of the Earth a The proper time rate in Eq. (7) is expressed with respect to local inertial RS, in which the gravitational field of external coordinate time t = t . ECI bodies can be represented in the form of tidal terms only The rate of coordinate time used in GNSS, however, is [7]. For GNSS it means that synchronization of the entire closely related to International Atomic Time (TAI), defined system of ground-based and orbiting clocks is performed in as the coordinate time at a surface of a constant effective the local inertial frame, or Earth Center Inertial (ECI) coor- gravitational equipotential at mean sea level in the ECEF. dinate system [2]. The coordinate time t of the ECI frame It is advantageous to exploit the fact that all clocks on the is defined at infinity, outside Earth’s gravity well [1 ]. If we reference surface tick at the same rate to redefine the rate of synchronize with respect to t = t , we are synchronizing ECI coordinate time by standard clocks at rest on Earth’s geoid. with respect to the coordinate time of an inertial RS, so that This is done by adopting the following coordinate change: the validity of transitivity can be assumed. dt = 1 + dt (8) Geoid ECI 3 The Proper Time Rate where t is the new coordinate time for clocks at rest on Geoid Earth’s geoid and  is the effective gravitational potential The relationship between the proper time  of a clock in the at the surface of the geoid (that also includes the Doppler vicinity of Earth and some coordinate time t is defined by term due to its rotation ) [2]. When this time scale change is the spacetime metric, which is derived from the metric ten- made, the proper time rate of Eq. (7) becomes: sor, the solution of the Einstein Field Equation. The metric V − d 0 v tensor completely characterizes the spacetime metric which = 1 + − . (9) 2 2 dt c 2c describes the relation between spatial and time coordinates. Geoid When Earth is modelled as a non-rotating spherical mass, We should remark here that in Eq. (9) we are comparing the the solution of such an equation leads to the Schwarzschild rate of a clock in the vicinity of Earth with respect to the metric. The Schwarzschild metric describes the region sur- time of a clock at rest on the surface of the geoid but with rounding a non-rotating spherical mass [17, 19]. 1 3 Relativistic Modelling for Accurate Time Transfer via Optical Inter-Satellite Links 281 Table 1 Values of used parameters [15] Parameter Notation Value −5 −1 Earth angular velocity 7.2921159 ×10 rad s −11 3 −1 −2 Universal gravitational G 6.67428 × 10 m kg s constant −1 Speed of light c 299792458 m s Earth mass M 5.972 × 10 kg WGS84 geoid semi-major a 6378136.6 m axis 2 −2 WGS84 effective potential  62636856.0 m s the understanding that synchronization is established in an underlying, locally inertial, reference frame (the ECI frame) [2]. The numerical value of the parameters and quantities presented in the preceding sections are given in Table 1. Fig. 2 Timing events: depicted are the trajectories of Satellite A and Satellite B with corresponding proper and coordinate times at signal transmission and reception in both directions of the communication 4 Autonomous Synchronization Some natural event, as an observation of occultation of a the following equation for the corresponding coordinate time star by a planet for instance, may be considered as a syn- t( ): chronizing event. Both observers may chose an arbitrary t( ) � � 2 initial condition linking their proper time to the coordinate V( (t )) −  v (t ) A 0 A −  = 1 + − dt (10) time, transform their proper time readings of observation of A A 2 2 c 2c the natural event into coordinate times t and t and then Ev Ev determine the coordinate offset  . A faster and more reli- AB Satellite A is thus able to transform its proper time of recep- able manner of synchronization is the artificial creation of tion  into the coordinate time instant t . If the trajectories A 3 a synchronizing event on the world line of the observer B of A and B are known with high accuracy, it is possible to via exchange of signals. This method of synchronization is determine the parameters  (t), v (t) and  (t), v (t) in the A A B B called “autonomous” or “independent” synchronization. In form of functions of the coordinate time scale t. Our goal is the system that we are considering, optical communication to retrieve the coordinate time of reception t so that we are enables the exchange of data with timestamps. This allows able to ascribe it to the reading  . The coordinate instant for time transfer between remote references [7]. t can be retrieved with the following iterative method [7]: 4.1 TwoW‑ay Synchronization (0) (0) t = t + t − t −  +  + (11) 0 3 0 Int hw 1 Prop Let’s consider a Two-Way Time Transfer (TWTT) scheme as presented in [7]: satellite A, located at  , transmits at A 1 (n) (n−1) t = t + t − t −  +  + (12) 0 3 0 Int hw 1 Prop coordinate time t a message with proper time stamp  . 0 A Satellite B receives it at coordinate time t , stamps it with proper time stamp  . After a delay satellite B re-transmits (n) t = lim t , n = 1, 2, 3, … 1 (13) it back stamping it with the proper time of re-transmission n→∞ . The signal is finally received at coordinate time t from B 3 where: Satellite A that stamps it at  . The coordinate propagation times for the round trip are respectively T = t − t and 2 AB 1 0 V( ()) −  v () B 0 B T = t − t . Assume that we have additional delays  and BA 3 2  = 1 − + d (14) Int 2 2 c 2c which denote the hardware delays relative to the reception (R) and transmission (T), with N = A, B . Figure 2 shows a model of this two-way exchange. Suppose we have an initial condition  (t )=  . Clock A A 0 A can transform its reading of any event happening at  solving 1 3 282 M. Dassié, G. Giorgi (n) (n) (n) 1 (n) (n) R R = T − T (15) (n) AB 0 AB Prop AB BA T = + AB 2 c c c � � (n) (23) ‖ (t )‖ + ‖ (t )‖ + R 2GM A 0 B AB (n) (n) E 1 R R + ln (n) AB (n) BA 3 (n) (16) T = , T = ‖ (t )‖ + ‖ (t )‖ − R AB BA A 0 B AB c c (0) Here, T is the roughly estimated propagation time that can AB (n) (n) R = ‖ (t )−  (t )‖ be computed thanks to the approximate knowledge of the (17) B A 0 AB 1 positions of the satellites. The first term on the right-hand side of Eq. (23) is the (n) (n) R = ‖ (t )−  (t +  )‖ (18) A 3 B Int BA 1 classical Euclidean time that light takes to go from point A to point B. The second and third terms are delays resulting 1 1 T R R T from the time dilation experienced by light when travel- =  −  +  − (19) hw A A B B 2 2 ling through a gravitational field and are purely relativistic effects. The third term is called the Shapiro delay. This Here,  is the integral of the proper time rate over the path Int expression for the propagation of light can be found by of clock B between reception from A and transmission from setting the spacetime element ds = 0 , solving for dt and B, which in coordinate time corresponds to t − t . The term 2 1 integrating it on the path from A to B. The two relativistic is the half difference of propagation time during the Prop terms are of the order of tens of picoseconds and therefore round trip between A and B (expressed with T and T AB BA (0) have to be taken into account to achieve picosecond-level respectively). The initial  can be estimated thanks to the Prop synchronization with an OWTT. Note also that reception approximate knowledge of the positions of the satellites. event t is determined with the knowledge of the trajecto- Finally,  identifies the residual hardware delay in a TWTT hw ries  (t),  (t) . In an OWTT, inaccurate knowledge of the A B exchange. This may be largely mitigated if the transmit and positions and trajectories would directly result in an offset receive delays at either side of the communication are simi- due to errors in the calculation of T . This is one of the AB lar. Satellite B observes the signal reception happening at main limitations of this method: before synchronization, . Thus, we can ascribe the calculated moment t to the B 1 even assuming that the satellite positions are known with observed moment  and, thereby synchronize clock B with cm accuracy, the resulting modelling errors are in the order clock A. We found the initial condition  (t )=  that B 1 B of hundreds of picoseconds in the determination of t (and allows clock B to transform all events seen in its timescale thus in a corresponding synchronization offset). Moreover, to the same coordinate time t as the one that clock A in an OWTT we can only achieve clock synchronization transforms to. if spurious terms, such as hardware delays in the optical terminals used to establish the link, can be modelled and removed. This may prove difficult: a characterisation of 4.2 Comparison with a OneW ‑ ay Synchronization these delays in a relevant operational environment would be difficult to achieve for picosecond-level accuracy. For completeness, we could consider a One-Way Time Due to the quasi-symmetry of the communication, a Transfer (OWTT), where we have a unidirectional TWTT is preferred because it mitigates all the problems exchange of signals from Satellite A to Satellite B. In of the OWTT mentioned above. Positioning errors, hard- Fig. 2 the communication would end at the signal recep- ware delays and additional relativistic offsets mitigate or tion from Satellite B at coordinate time instant t . In this cancel out in the expressions for determining t . I n de e d , case t can be determined with the following iterative in a TWTT we are interested in characterizing the differ - method: ence of propagation times during the round trip and not (0) (0) their absolute magnitude. If we compute Eq. (15) using the t = t + T (20) 1 AB expression for the one-way propagation times (23) we can see that the difference of the additional relativistic terms (n) (n−1) t = t + T (21) is of the order of femtoseconds and is therefore negligible. 1 AB In fact in Eq. (16) the expression for the propagation time (n) is simpler and only includes the Euclidean term. It will be t = lim t , n = 1, 2, 3, … (22) n→∞ shown in a dedicated section (Sect. 6.2.2) that positioning errors are also mitigated in a TWTT. where: 1 3 Relativistic Modelling for Accurate Time Transfer via Optical Inter-Satellite Links 283 n n max 4.3 System Synchronization GM V (r, 𝛼 , 𝛽 ) =− 1 + P (sin 𝛽 ) E nm r r n=2 m=0 One can synchronize any arbitrary number of other satellites (25) using a cascaded approach, i.e. Satellite A gets synchronized ̄ ̄ × C cos(m𝛼 )+ S sin(m𝛼 ) nm nm with B via a time transfer, B gets synchronized with C and so on until all satellites are synchronized to the same coor- where: dinate scale. However, this implies that at least one clock exchanges signals with a clock keeping the coordinate time. • a is the semi-major axis of the WGS 84 Ellipsoid; In a GNSS this last clock would be on Earth. Essentially, ̄ ̄ • C and S are the normalized gravitational coefficients nm nm only one ground station would be sufficient to guarantee the contained in EGM2008; link with the terrestrial time scale. The presented synchro- • P is the normalized associated Legendre function of nm nization method allows for a significant degree of auton- degree n and order m; omy of the satellite system. In the extreme case, lacking and  are respectively the longitude and latitude of the even the ground station, the satellite clocks realize a “space satellite position in an ECEF frame. clock” whose time scale definition is not affected by the poor knowledge of the geopotential on the surface of Earth Not all terms in the summation are significant: the model [20]. Such a clock, moving in an unperturbed orbit and well- should only retain n terms, sufficient to guarantee that the max defined coordinate system, would bring significant value and model inaccuracies remain below ps-level. The term V is SC new capabilities to time and frequency transfer worldwide, the gravitational contribution of body C (mass M ) situated to precision geodesy and terrestrial reference frames, to at  on satellite S situated at  . This is expressed as [20]: earth-environmental science and to navigation systems [3]. 1 1 V =−GM − − (26) SC C r r SC C 5 Picosecond Synchronization Models C where  =  −  and r = ‖‖. In both time transfer methods presented, in order to reach The relation between the rate of a clock in the near Earth picosecond precision it is necessary to have precise models region (proper time  ) with respect to coordinate time on of the proper time rate and of the propagation time T AB dt Earth’s geoid becomes: (or the difference of propagation times in a TWTT). V  V d v E 0 SC ≈ 1 + − − + (27) 2 2 2 2 dt c c 2c c 5.1 T he proper Time Integral C≠E We recall here the expression of the proper time rate of a 5.2 Evaluation of Proper Time Rate Terms for Kepler clock in Earth’s vicinity with respect to TAI coordinate time Satellites t: V((t)) − d(t) v(t) In this section, we are interested in determining the number = 1 + − (24) 2 2 dt c 2c of terms to be included in the proper time rate expression to be integrated in order for the integral terms to meet dt Integrating Eq. (24) it is possible to transform any reading of the requirement of picosecond-level accuracy. We need to the clock  to events of coordinate time scale t. This integral characterize which effects play a role within the time of a has to be solved analytically or numerically in order to relate signal exchange, which in the case of typical GNSS satellites instants of the two scales. The complexity of finding a solu- in Medium Earth Orbit (MEO) is always within a second. tion depends on the model of the potential. The gravitational The Kepler constellation consists of two segments: a set of potential V in Eq. (24) is the sum of the Earth’s gravitational navigation satellites in MEO, assigned approximately to the potential V and the one created by the cumulative contribute same orbital slots of the current Galileo constellation, and a of other celestial planets, V . SC smaller set of satellites in upper Low Earth Orbit (LEO). The The Earth’s gravitational potential V is defined through relevant orbital parameters are summarized in Table 2. a summation of spherical harmonics [14]: A list of coefficients, degrees and orders, with up to 2190 degrees, can be found at [13]. 1 3 284 M. Dassié, G. Giorgi Table 2 Summary of main Kepler parameters Table 3 Number of geopotential harmonic terms to take into account in the proper time rate for different thresholds Segment Type Inclination Semi-major axis Constellation Threshold [s/s] n max MEO Walker 24/3/1 29601.3 km −12 ◦ LEO 2 (3 LEO Walker 6/2/1 7626.3 km 89.7219 recom- mended) −15 −18 −12 MEO 2 −15 −18 10 5 We can see in Fig.  3 that only two terms (quadrupole formulation of the potential) are sufficient to describe the proper time rate of satellites in both constellations at the −12 order of 10 . From the third term the information apported can be considered negligible. In any case it is clearly visible that the geopotential term of order n = 2 for LEO satellites is very close to this threshold. In order to keep some margin it is advisable to consider a potential expansion with n = 3 max for the LEO constellation. As expected more terms in the Fig. 3 Maximum contribution of Earth’s potential harmonic terms of potential expansion are needed for stricter thresholds. A degree n on MEO and LEO satellites proper time rates summary of the results is presented in Table 3. Using the orbital simulation data from GMAT each term The satellites of both segments are homogeneously dis- of Eq. (27) for every single satellite has been computed and the maximal contribution of each term has been recorded. tributed over their orbital planes, i.e. 45 difference in mean anomaly for MEO and 60 for LEO neighbour satellites. The orbital parameters of the sun, moon and other planets have been artificially modified to simulate the case where The planes are also homogeneously distributed around the globe with a 120 separation in the right ascension of the every other body considered is at its closest approach during the whole simulation. This is an extreme case that is very ascending node (RAAN) for MEO planes and 90 for LEO planes. The argument of perigee is 0 for both segments. unlikely to happen but it gives an upper bound to the shift caused by each one of those bodies. Since the communication time is always shorter than 1 s we can directly look for terms in the proper time rate that are Figure 4 shows the magnitude of the maximal contribu- −12 tions of the terms in Eq. (27) for MEO and LEO satellites smaller than the threshold of 10 s/s. For completeness and future use we evaluate all terms of the proper time rate in the constellation. For picosecond accuracy only the geo- −12 −15 potential term and the second order Doppler term need to be for different satellites against 3 thresholds: 10 , 10 and −18 10 s/s. The satellites’ orbits are simulated using the open considered. Gravitational effects of external bodies need to −15 be taken into account for the stricter thresholds of 10 s/s source software General Mission Analysis Tool (GMAT), −18 available from NASA [12]. The simulation interval is chosen (Moon influence on MEO satellites) and 10 s/s (Moon and Sun for both MEO and LEO satellites). Table 4 summarizes to be the 10 days and the integration step to propagate the orbits is 30 s. all the terms to be taken into account for a given constella- tion and threshold. We have presented in Eq. (27) the terms to consider for a full model of the proper time rate for a clock located at (t) in the vicinity of the Earth with respect to TAI coordi- nate time. As presented in Eq. (25), the “geopotential” V 6 Impact of Orbit Parameter Errors (gravitational potential of Earth) is defined through a sum- mation of spherical harmonics. We name each term of the We are now interested in the sensitivity of the derived mod- geopotential V . Figure 3 shows the maximal contribution of the single els to orbit parameter errors. To analyse it we evaluate the resulting errors in proper time rate and in the difference of geopotential terms of degree n manifested during the simulated 10 days. propagation times arising from errors in position, range and 1 3 Relativistic Modelling for Accurate Time Transfer via Optical Inter-Satellite Links 285 Table 5 Allowed velocity error Orbit H  [m/s] to remain below a proper time rate error threshold H −12 LEO 12.42 −15 −3 10 12.42 × 10 −18 −6 10 12.43 × 10 −12 MEO 10 24.40 −3 −15 24.48 × 10 −18 −6 10 24.48 × 10 (v + ) d d 0 0 (30) − = − = H 2 2 dt dt 2c 2c w/Error w/o Error which leads to: = v + 2Hc − v (31) Fig. 4 Maximum value of terms in the proper time rate expression for Assuming circular orbits, the orbital velocity of a body is LEO and MEO satellites during the simulated 10 days GM v = where a is the orbit radius. Table 5 summarizes the results for all thresholds H and both types of satellites Table 4 Terms to take into account in the proper time rate for differ - considered for the Kepler constellation. It is noticeable how ent thresholds the requirement for the uncertainty in the velocity is techno- Constellation Threshold [s/s] Terms −12 logically not hard to meet (at least for the 10 case). The −12 velocity can be determined with a higher level of precision LEO 10 Geopot, Doppler without major difficulties. −15 10 Geopot, Doppler −18 Geopot, Doppler, Moon, Sun −12 6.2 Position Error Margin MEO Geopot, Doppler −15 Geopot, Doppler, Moon 6.2.1 Impact of Position Errors on the Proper Time Rate −18 Geopot, Doppler, Moon, Sun Satellite position errors impact the determination of the proper time rate because of the dependence of the Earth’s velocity. In fact we want to determine the permitted error on geopotential on position. Particularly affected is the uni - those parameters that leads to: form geopotential term, which is the dominant term in the d d expansion and depends on the norm of the position vector − = H [s∕s] (28) (t) . Consider (r + ) the norm of the position vector of the dt dt w/Error w/o Error 0 satellite with error and r the same quantity without error: � � −  = H [s] � � V V Prop Prop �w/Error �w/o Error − = H (32) 2 2 c c w/Error w/o Error ⎧ −12 (29) ⎪ GM −15 Using the assumption of uniform gravity V(r)= − : where H = 10 . −18 ⎩ GM 1 1 − + = H (33) c r +  r o o which leads to: 6.1 Velocity Error Margin −1 Hc 1 = − + − r (34) An error in the velocity of the satellites would impact the GM r E 0 determination of the proper time rate, affecting the second- Table 6 summarizes the results for all thresholds H evaluat- order Doppler term. Consider (v + ) the velocity of the ing Eq. (34) for both types of satellites of the Kepler con- satellite with error and v the velocity without error: stellation. It can be seen again how the requirement for the 1 3 286 M. Dassié, G. Giorgi Table 6 Permitted radial Table 7 Clock offsets resulting from 4 m positioning errors in TWTT Orbit H  [m] position error to remain below a assuming that those errors remain constants during the whole signal proper time rate error threshold −12 3 exchange process LEO 10 13.136 × 10 −15 13.1149 Communication t − t  [s] |H| [s] 2 0 −18 −3 10 13.114 × 10 −18 MEO-MEO neighbours 0 9.8986 × 10 −12 3 MEO 10 198.900 × 10 −13 0.1 1.6545 × 10 −15 10 197.573 −13 0.5 8.2728 × 10 −18 −3 10 197.572 × 10 −17 MEO-MEO next-neighbours 0 1.2348 × 10 −13 0.1 1.6544 × 10 −13 0.5 8.2723 × 10 −12 uncertainty in the positioning is very loose (for the 10 s/s −13 LEO-MEO 0 3.4792 × 10 case). There is no difficulty in knowing the radial position −13 0.1 3.2474 × 10 with such precision. −13 0.5 3.4630 × 10 6.2.2 Impact of Position Errors on the Propagation Time The offset is minimal when  ≈− . This is the case for The real impact of satellite position errors manifests itself in AB BA an approximately simultaneous exchange of signals, where the calculation of the difference in propagation times, rather t ≈ t . In practice, since the clocks will be roughly synchro- than on the proper time rate. Assume that the real positions 0 2 nized, both Satellite A and Satellite B will send a signal at (t) and  (t) are affected by positioning errors (in vectorial A B proper times  and  which they can approximately relate form) expressed respectively as  (t) and  (t) due to uncer- A0 B2 A B to coordinate times t ≈ t . tainties in the orbit determination. The difference in propaga- 0 2 The worst case scenario is when  =− and both tion times is then: A B uncertainties are parallel to  +  . Preliminary analy- AB BA ‖ +  (t )−  (t )‖ � AB B 1 A 0 sis of Precise Orbit Determination (POD) for the Kepler Prop �w/Error 2c constellation shows that by considering a simultaneous (35) ‖ +  (t )−  (t )‖ BA A 3 B 2 application of a large number of modelling errors (with- 2c out using error reduction techniques), satellite 3D posi- tion errors are expected to be smaller than 70 cm. In the As a first approximation we can assume that the posi- mentioned case, POD is performed by a single ground tion errors remain basically constant during the com- station with the support of the LEO segment, without any munication period which is expected to last a few hun- assumption of prior synchronization or use of ISLs [9]. In dreds of ms, i.e.  ∶=  (t )≈  (t )≈ const. A A 0 A 3 an operational scenario, satellites would infer their posi- and  ∶=  (t )≈  (t )≈ const. We have that B B 1 B 2 tion from the broadcast ephemerides based on predicted ‖ −  ‖ ≪ ‖ ‖ so we can use the Taylor expansion B A AB orbits. The accuracy of the predicted orbits is expected to of Eq. (35). be lower than the one of POD orbits but it is nevertheless Thus: expected to be much better than 1 meter. To remain con- R R servative, we assume here that the positions of all com- AB BA ≈ − Prop municating Kepler satellites are known with a relatively w/Error c c (36) large uncertainty ‖ ‖ = ‖ ‖ = 4 m. A simulation has A B +   − AB BA B A been performed by injecting orbital errors at this level and 2c the results are given in Table 7. In the simulation different types of transmission delays t − t are evaluated. In the 2 0 =∶   + H first case optical signals are transmitted simultaneously, (37) Prop w/o Error i.e. t = t . In the other cases Satellite B transmits after 0 2 where H corresponds to the last term on the right hand side delay t − t with respect to the moment of transmission 2 0 AB BA of Eq. (36),  = and  = . t from Satellite A. We can see from the results that even AB BA 0 ‖ ‖ ‖ ‖ AB BA when a large uncertainty in the positions of the satellites We identify here a synchronization offset H due to uncer- is considered and the communication is very asymmetric tainty in orbital position determination, expressed as: ( t > t ), we can still model relativistic effects to within 1 2 0 +   −  ps accuracy. This means that with the current orbit deter- AB BA B A (38) H = mination capabilities we can accurately model relativistic 2c effects to below 1 ps. 1 3 Relativistic Modelling for Accurate Time Transfer via Optical Inter-Satellite Links 287 and 3 we have defined simultaneity and coordinate syn- 7 Technological State of the Art chronization in the relativistic domain. A synchronization method based on a Two-Way Time Transfer was presented So far, time transfer has been presented in abstract form in Sect.  4. Then the relativistic effects to be taken into through a mathematical description of the process. The goal account for a time transfer with the necessary degree of of this section is to provide an overview of the technology accuracy have been studied, and Sect.  5 shows that the required to make the presented synchronization methods a gravitational influence of other celestial bodies can be reality. Two-way time transfer at ps-level is possible via com- neglected and only the Earth geopotential has to be taken munication systems based on lasers, already space-qualified into account. The Earth’s geopotential needs to be char- and operative for commercial satellite data relay applications acterised with a higher degree of accuracy than in cur- [5]. Early ground-to-space tests of time transfer via non-coher- rent GNSS systems, considering harmonics beyond the J ent optical links have already demonstrated accuracies in the moment. For a correct relativistic modelling with the goal range of hundreds picoseconds. Examples of such tests are of picosecond-level synchronization accuracy, there is a the Laser Time Transfer technology validated at 300 ps-level need to keep the errors on orbits below a certain threshold. on Beidou satellites [8] and the Time Transfer by Laser Link These thresholds have been calculated and presented in (T2L2), which demonstrated time transfer between a number Sect. 6. It was discussed how a Two-Way Time Transfer of International Laser Ranging Service (ILRS) stations and the picosecond synchronization is not only fundamentally fea- Jason-2 satellite below the 100 ps-level [4]. These methods use sible, but can also be implemented in practice. non-coherent links. By exploiting laser-based coherent links, a A relativistic analysis is a fundamental step in the defi- network of two or more frequency references synchronized at nition of the GNSS processing architecture, since in this sub-ps level can be established. A DLR laboratory demonstra- framework synchronization offsets derive from fundamen- tor of coherent links is being developed to verify optical range, tal characteristics of nature. The feasibility of relativistic time transfer, and data transmission. The demonstrator consists synchronization with OISLs confirms the possibility of of two terminals, running a bidirectional free-space optical separating space and time in the error estimation process, link in the laboratory, with single-mode fiber coupling in the resulting in a much higher level of synchronization across receivers at both sites. The optical carrier is modulated by a the constellation than is achievable in current architectures. fast ranging sequence and a slower data stream. The coherent This has the potential to greatly improve accurate orbit transceivers enable time-stamping received reference bits with determination and consequently the navigation service for sub-ps precision, enabling the exchange of measured time- end users. Furthermore, the significant degree of autonomy of-arrival information between the paired satellites, which in synchronization enabled by the proposed TWTT scheme in turns enable two-way time transfer to retrieve clock off- has a potentially large impact on infrastructure and opera- sets and perform inter-satellite ranging at sub-mm level [18]. tional costs. The feasibility of an innovative concept of a Such optical transceivers could be integrated in existing opti- fully autonomous ’space clock’ was also addressed. The idea cal communication terminals, such as those employed in the of such a clock could be further investigated and has the SpaceDataHighway operated in space since a few years in the potential to strongly impact the the field of time metrology, framework of the Copernicus program to optically link LEO with the definition of a spatial time scale less affected by and GEO satellites. This integration is the focus of a DLR various terrestrial phenomena. mission named COMPASSO, initiated in 2021, aiming at test- ing a coherent optical link for time/frequency transfer, rang- Acknowledgements I would like to thank my advisor Dr. Gabriele ing and data exchange between a terminal on the international Giorgi for his excellent supervision both during the thesis and after- space station and an optical ground station in Europe. The wards. It is thanks to his support that this work has seen the light of day. mission will hopefully pave the way to further validation mis- In addition to my advisor, I would like to thank the director of the DLR Institute of Communications and Navigation, Christoph Günther, for sions aimed at testing autonomous accurate synchronization the support and nice discussion regarding this work. I would also like to between satellites in higher orbits, and future utilization in thank my parents and aunt, who have always supported me throughout GNSS constellations to augment navigation solution accuracy, my academic career and my life in general. system operability and robustness of operations. Funding Open Access funding enabled and organized by Projekt DEAL. This work was done in the context of the ADVANTAGE pro- ject, supported by the Helmholtz-Gemeinschaft Deutscher Forschun- 8 Conclusion and Future Perspectives gszentren e.V. (Helmholtz Association of German Research Centers) under grant number ZT-0007, and in the OTTEx project, supported This work focused on analysing how to accurately model by the European Space Agency (grant number H2020-ESA-038.04). relativistic effects to enable ps-accurate time transfer Availability of Data and Material Not applicable. between satellites in MEO and LEO orbits. In Sects. 2 1 3 288 M. Dassié, G. Giorgi Code availability Custom code. and experiment of onboard laser time transfer in chinese beidou navigation satellites. Adv. Space Res. 51(6), 951–958 (2013). https:// doi. org/ 10. 1016/j. asr. 2012. 08. 007 Declarations 9. Michalak, G., Glaser, S., Neumayer, K., König, R.: Precise orbit and earth parameter determination supported by leo satellites, Conflict of interest On behalf of all authors, the corresponding author inter-satellite links and synchronized clocks of a future gnss. Adv. states that there is no conflict of interest. Space Res. (2021) 10. 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Journal

Aerotecnica Missili & Spazio – Springer Journals

Published: Sep 1, 2021

Keywords: Time transfer; Proper time rate; Intersatellite links; Relativistic propagation

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