## 3 Construction of Mathematical Model
### BDS-Based Interplanetary Link for the Irradiation of Non-Cooperative Spacecraft in Earth-Moon Space  
This subsection will focus on the establishment of the transmission equations of inter-satellite link signals in cislunar space for GNSS satellites located in different orbits, the transmission of inter-satellite link signals reflected by non-cooperating targets, the reception equations at the receiving end of the CEI array, and the inter-satellite link transmission loss equations, as described separately below.  
2.1 The Transmission Equations of the Inter-Satellite Link Signals of Beidou-3 Satellites in Different Orbits in Cislunar Space  
Based on the inter-satellite link signals of BDS satellites, the pseudocode ranging equations between BDS satellites located in different orbits and non-cooperative targets in cislunar space are shown below.  
\[\begin{array}{l}D_{(iBD\to space)}=\left|\mathbf{r}_{space}(t_{2})- \mathbf{r}_{iBD}(t_{i1})\right|+c\cdot\delta t_{space}(t_{i2})\\ -c\cdot\delta t_{iBD}(t_{i1})+c\cdot\tau_{iBD}+e_{i(BD\to space)}+n_{i(BD \to space)}\end{array} \tag{9}\]  
where \(i=1\), \(i=2\), \(i=3\) is the MEO, GEO, and IGSO orbit, respectively; \(t_{i1}\) is the moment when the BDS satellite launches the interplanetary link, and \(t_{i2}\) is the moment when the interplanetary link reaches the targets; \(\mathbf{r}_{iBD}\) and \(\mathbf{r}_{space}\) is the position vector of BDS satellites and targets at different orbital altitudes in the geocentric inertial coordinate system; \(\delta t_{iBD}\) and \(\delta t_{space}\) are the clock differences of BDS satellites and targets relative to BDS system time (BDT), respectively; \(\tau_{iBD}\) is the hardware delay of BDS satellites at the time of launching the interplanetary link; \(e_{i(BD\to space)}\) is the other errors to be corrected, including antenna phase center shift, relativistic effects, etc.; and \(n_{i(BD\to space)}\) is the observation noise.  
Figure 5: The dynamical constraint of non-cooperative spacecraft in cislunar space.  
2.2 Transmission of Interplanetary Link Signals Reflected by Non-Cooperative Spacecraft in Cislunar Space and the Final Reception Equation at the Receiving End of the CEI Array  
Imitating the above expression, we can obtain the pseudo-code ranging equation between the reflection of the BDS satellite's inter-satellite link signal after shining on the targets and the ground CEI receiver array, as is shown below.  
\[\begin{array}{l}D_{i(space\to CEI)}=\left|\mathbf{r}_{CEI}(t_{i3})- \mathbf{r}_{space}(t_{i2})\right|\\ +c\cdot\delta t_{CEI}(t_{i3})\\ -c\cdot\delta t_{space}(t_{i2})\\ +c\cdot\tau_{iCEI}+e_{i(space\to CEI)}+n_{i(space\to CEI)}\end{array} \tag{10}\]  
where \(t_{i3}\) is the moment when the interstellar link reflected by the non-cooperative spacecraft reaches the CEI array on the ground; \(\mathbf{r}_{CEI}\) is the position vector of the ground-based CEI observation array in the geocentric inertial coordinate system; \(\delta t_{CEI}\) is the clock difference of the CEI observation array on the ground relative to the BDT; \(e_{i(space\to CEI)}\) is the other errors to be corrected, including atmospheric delay errors (mainly ionospheric and tropospheric delay errors), relativistic effects, station antenna phase center shifts, tidal effects, etc.; and \(n_{i(space\to CEI)}\) is the observation noise.