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

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 (synchronization) 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 accuracy. Also, we show how to practically achieve autonomous synchronization via a sequence of pair-wise synchronizations across all satellites of the constellation.

Autonomous Satellite System Synchronization Schemes via Optical Two-Way Time Transfer and Distributed Composite Clock

“…In the specific case of MEO and LEO satellites, all bodies other than Earth account for much less than picosecond per second and can be neglected. Nonetheless, the geopotential has to be expressed as a sum of multipole terms up to the third order (Dassié & Giorgi, 2021).…”

“…An incorrect knowledge of position and velocity of the satellite affects the proper-to-coordinate transformation. As shown in (Dassié & Giorgi, 2021), the error margins are generous: for integrals over 1 s, we are still able to transform the satellite clocks timestamps into coordinate time instants with picosecond-level accuracy even with position errors of the order of thousands of meters and velocity errors in the order of meters per second.…”

“…Reordering the term in (7) one can see that the propagation times appear in the form of difference 1 2 (𝑇 𝐵𝐴 − 𝑇 𝐴𝐵 ), so one can use a simplified version of (3) where only the Euclidean term is considered for both 𝑇 𝐴𝐵 and 𝑇 𝐵𝐴 . The difference of Shapiro delays in the roundtrip is much lower than a picosecond and can be neglected (Dassié & Giorgi, 2021). An error in the determination of the position of a satellite would impact the estimation of the difference in the roundtrip propagation time and consequently result in a biased estimate of the inter-satellite clock offset.…”

“…The first term of ( 24) is an offset due to a constant error present at the instant of the first transmission. It was shown in (Dassié & Giorgi, 2021) that meter-level position errors still allow this term to remain in the sub-picosecond level. The remaining terms in ( 24) are offsets arising from the growth of the error during the exchange.…”

“…If such effects are not properly accounted for, relativistic biases will jeopardize the synchronization system (Ashby, 2003). In a previous work (Dassié & Giorgi, 2021), we presented and analyzed a relativistic two-way synchronization scheme based on the exchange of time stamps via optical signals, which allows estimating the offset between two satellite clocks with respect to a defined coordinate time scale. The model therein guarantees a time transfer at picosecond-level.…”