# LEO Precise Orbit Determination with Inter-satellite Links
## 2 ISL Observation Model  
In principle, two satellites can establish an ISL as long as they are visible to each other [35]. A pair of satellites (i.e., satellite \(A\) and \(B\)) and the Earth are shown in the Figure 1, with O being the center of the Earth and R being the radius of the Earth. Our definition of \"visible\" for satellite \(A\) and \(B\) is formulated as follow:  
\[\left| \begin{array}{c}\overset{\rightarrow}{OA}\times\overset{\rightarrow}{ OB}\\ \overset{\rightarrow}{OA}-\overset{\rightarrow}{OB}\end{array} \right|\geq R \tag{1}\]  
In other words, satellite \(A\) and \(B\) are considered to have the capacity of an ISL if Equation (1) can be satisfied. It is worth mentioning that in practical application the direction of satellite antenna also impose restriction on ISL. Taking BDS-3 as an example, a phased array antenna switches the beam direction during different timeslots in order to point to different satellites, and the pointing limit is (\(-60^{\circ}\), \(60^{\circ}\)) [36]. Since all our work are based on simulated constellation, we neglect the effect of satellite antenna direction for simplicity. The ISL ranging is a dual one-way observation mode with a pair of satellites receiving the signal from each other almost simultaneously, and it is only capable of pseudo-code measurements. Figure 2 vividly depicts the ISL mechanism between satellite \(A\) and \(B\). Given that satellite \(B\) receives the pseudo-code measurement \(\overline{\rho}_{A}^{B}(t_{A})\) from satellite \(A\) at its local time \(t_{A}\) while satellite \(A\) receives the pseudo-code measurement \(\overline{\rho}_{B}^{B}(t_{B})\) from satellite \(B\) at its local time \(t_{B}\) and the forward and backward signal transmitting time is \(\Delta t_{AB}\) and \(\Delta t_{BA}\), respectively. Then, the dual pseudo-code measurements can be expressed as follows:  
\[\overline{\rho}_{A}^{B}(t_{A})=\rho_{A}^{B}(t_{A}+\Delta t_{AB})+\delta_{B}(t_ {A}+\Delta t_{AB})-\delta_{A}(t_{A})+l_{B}^{R}-l_{A}^{S}+O_{AB}+\varepsilon_{BA} \tag{2}\]  
\[\overline{\rho}_{B}^{A}(t_{B})=\rho_{B}^{A}(t_{B}+\Delta t_{BA})+\delta_{A}(t_ {B}+\Delta t_{BA})-\delta_{B}(t_{B})+l_{A}^{R}-l_{B}^{S}+O_{BA}+\varepsilon_{BA} \tag{3}\]  
where \(\overline{\rho}_{A}^{B}(t_{A})\) and \(\overline{\rho}_{B}^{A}(t_{B})\) are the measured ranges at signal receiving epoch of each satellite and \(\rho_{A}^{B}=c*\Delta t_{AB}\) and \(\rho_{B}^{A}=c*\Delta t_{AB}\) (\(c\) is the speed of light travelling in vacuum space, which is about 299,792.458 km/s) are the forward and backward signal space propagation (i.e., the real ranges between satellites at their signal receiving epoch), respectively; \(\delta_{A}\) and \(\delta_{B}\) represent the clock bias of each satellite; \(l_{A}^{S}\) and \(l_{B}^{S}\) denote the hardware delay of satellite \(A\) and \(B\) at the sending end while \(l_{A}^{R}\) and \(l_{B}^{R}\) at the receiving end; \(O_{AB}\) and \(O_{BA}\) are phase center offsets (PCOs), phase center variations (PCVs), relativistic effects, and some other corrections; \(\varepsilon_{AB}\) and \(\varepsilon_{BA}\) are the corresponding measurement noise. It is worth noting that the abovementioned ISL model is defined for GNSS constellations. Since there are currently no LEO satellites with ISL loads, it is difficult to analyze the error factors affecting the inter-satellite ranging under LEO conditions. So we use the GNSS ISL observation model as substitution for the LEO ISL. The forward and backward ISL observations are nearly at the same time but still have differences. In the BDS-3 ISL observations, the time difference is less than 3 s [28]. So at first, what we need is to transform the dual measurements to the same time \(t_{0}\) with the auxiliaries of broadcast orbits and clock biases. This procedure is defined as epoch naturalization, during which all the hardware delays are considered to be constant. Figure 1: A pair of visible satellites.