## 1 Introduction  
Some existing satellite time synchronization systems, such as the gravity recovery and climate experiment (GRACE) achieve synchronization by means of compensating the clock errors at the ground station rather than producing the synchronized and syntonized timing signals [1, 2]. Achieving real-time synchronization and syntonization of satellite systems should produce more ideal signals and eliminate the various errors due to the inaccuracy of the on-board clocks. In general, atomic clocks have better long-term stability but also worse short-term stability, greater volume, weight, power, price and shorter lifetime compared with a high quality crystal oscillator (although chip-scale atomic clocks have been developed, their stability is not as good as full-scale atomic clocks and they are not space qualified). For this reason, global navigation satellite systems (GNSS), such as the global positioning system (GPS), adopts a time-keeping system (TKS) that couples the atomic clock with a voltage-controlled crystal oscillator (VCXO) on-board the satellite to produce the reference timing signal [3, 4]. This sophisticated and expensive military-based system is not suitable when low cost and low complexity is required. The Japanese Quasi-Zenith Satellite System proposed a solution that they refer to as \"remote time synchronization system for the on-board crystal oscillator\" (RESSOX) to realize time synchronization between the ground station and satellites [5, 6]. Different from GPS, the QZS employs a VCXO as the on-board clock in the RESSOX scheme. This VCXO is remotely steered by an atomic clock that is located in the ground station by means of a series of feed-forward control and feedback control. The positioning accuracy of RESSOX is better than using GPS & QZSS with on-board atomic clock [7]. However, it was not implemented in the final QZSS owing to the budget constraints. The systems mentioned above are steered by ground stations, which means the redundant ground stations and personnel are essential. Autonomous time synchronization for space systems is put forward in recent years. In [8], the NAMURU V3.2 spaceborne receiver, which is developed specifically for CubeSat formation flying, is disciplined to an external reference-GPS time. However, this kind of one-way time dissemination method has limited precision (20 nanoseconds). Hence starting from Block IIR, GPS adopts inter-satellite links combined with the polling time division duplex scheme (PTDD) to achieve autonomous time synchronization in cases where the ground station is not available [9, 10]. Similar discussions about other GNSS systems could be found in [11, 12], although these are not actually being implemented yet. In addition, some other scientific-based space missions are also being deployed, In [13], a method based on asynchronous two way time-stamping exchange is being employed, where the clock skews and clock offsets are estimated but not adjusted. Based on the above discussions, when spacecraft activities require autonomous, robust, high-accuracy time synchronization performance, the question regarding how to reduce overall satellite cost, power consumption, on-board weight and volume, and improve satellite lifespan becomes an interesting issue. In this paper, we propose a remote physical time synchronization system based on a VC-OCXO. The main contributions of this paper are threefold. First, it has the STDD & FDD configuration, in which the satellites simultaneously communicate with each other via different frequencies. Different from the PTDD adopted by GPS crosslink, STDD has the advantages on ranging precision, clock synchronization precision, ranging efficiency and channel utilization ratio [14]. Second, a functional model for the synchronization loop that contains clock error measurement, prediction and adjustment, is designed to be applicable to various space scenarios. A relative motion compensation method based on the proposed model is provided for this purpose.