## 3 pLEO Time Scale Development
### Inter-Platform Measurements Enabled by High-Precision Time Synchronization  
In the framework of the clock ensemble principle with measurements, one begins by comparing the orbiting clocks to one of the ensemble clocks using precise clock-to-clock difference measurements. As illustrated in Figure 6, these measurements are centered on understanding the differences between the initial states of two ensemble clocks. These crucial measurements are activated by events linked to the holdover performance of the physical clocks. They are subsequently processed through a Kalman filter, complemented by a covariance reduction extension to effectively address the unobservability challenge presented by a system of \(N\) independent clocks. It is essential to recognize that, in reality, only \(N-1\) of these clocks can be independently observed from the measurements obtained. When implementing a clock ensemble, the inputs to the clock-ensemble Kalman filter are clock-to-clock difference measurements from each bi-directional radio link. For instance, differential phase measurements output from each platform between onboard \(i\)th and \(j\)th clocks are given by  
\[z_{ij}(t_{k})=p_{i}(t_{k})-p_{j}(t_{k})+v_{ij}(t_{k})\,, \tag{36}\]  
where \(v_{ij}(t_{k})\) is a zero-mean Gaussian measurement noise with its covariance adjusted for the behavior of the difference between clocks \(i\) and \(j\). Figure 6: Inter-platform measurements. Understanding the complexities of clock measurements is essential as they are significantly influenced by platform dynamics and relativistic effects. A prime example of this is the Sagnac effect, which describes the phase shift that can occur between two measurements taken in opposite directions along the same closed path [20]. In pLEO scenarios, accurately synchronizing time across different platforms requires accounting for the Sagnac effect to ensure precise differential phase measurements. The research presented in [21] introduces an Enhanced Multi-Way Time Transfer (EM-WaTT) solution, which is specifically designed to address the challenges associated with dynamic time coordination. This approach is particularly important when inter-satellite link delays are not uniform across different platforms. As shown in Figure 7, the EM-WaTT method consists of three strategic steps, assuming that the secondary platform, identified as platform \(j\), and the primary platform, referred to as platform \(i\), can exchange timing information through wireless inter-platform communications. By implementing this innovative solution, greater accuracy in time synchronization across various platforms can be achieved. Step 1. The secondary platform, \(j\), launches its time synchronization protocol by sending a message included with the transmit time stamp, \(t_{TX}^{S}\) to the primary node, \(i\). Upon receiving the message, the primary platform, \(i\), will add the receive timestamp, \(t_{RX}^{M}\) on the primary clock to the message. Step 2. The primary platform, \(i\), immediately sends the updated message back to the secondary platform, \(j\), which adds the receive time stamp, \(t_{RX}^{S}\) on the secondary clock to the received message. Step 3. The secondary platform, \(j\), will then perform the clock adjustment based on the updated message, i.e.,  
\[t_{RX}^{M}-t_{TX}^{S} =\frac{R_{1}}{c}+T_{RX}^{M}+T_{TX}^{S}+\Delta t \tag{37}\] \[t_{RX}^{S}-t_{TX}^{M} =\frac{R_{2}}{c}+T_{TX}^{M}+T_{RX}^{S}-\Delta t\,, \tag{38}\]  
where in this context, \(R_{1}\) and \(R_{2}\) represent the distances between the secondary and primary platforms at the time of transmission, respectively. The terms \(T_{TX}^{S}\) and \(T_{RX}^{S}\) refer to the transmit and receive processing times of the secondary platform \(j\). Similarly, \(T_{RX}^{M}\) and \(T_{TX}^{M}\) indicate the processing times to receive and transmit from the primary platform \(i\). In practical applications, these processing times are fixed based on local signal processing clocks with a limited number of clock cycles.