# Parameter Design for Inter-Satellite Laser Link Acquisition Under Weak Light Conditions
## 2 Acquisition System Influencing Factors  
The strategy for establishing laser links in the Taiji program is sequential, meaning we establish one laser link at a time. For now, we assume that the signal acquisition process on the second and third arms will theoretically mirror that of the first arm once its laser link is established [12]. Consequently, we focus solely on the construction process of the first laser link.  
The overall process of acquisition of one link is illustrated in Figure 1 and described as follows: first, one of the two SCs on the arm, referred to as the scanning SC, emits a scanning beam. This beam scans the entire UC along a predetermined trajectory. When the scanning beam passes over the other SC, referred to as the receiving SC, the acquisition camera carried by the receiving SC detects a sufficiently strong spot signal within a set integration time of the acquisition camera. The attitude difference between the scanning beam and the receiving SC is determined by measuring the difference between the spot's center and the pre-calibrated reference position. The receiving SC then adjusts its own attitude based on this relative attitude information to align the laser spot position with the reference spot position. The receiving SC then emits a beam aimed at the scanning SC. Similarly, the acquisition camera on the scanning SC detects the laser beam and then adjusts its attitude accordingly. Through continuous adjustments, the two SC eventually achieve mutual alignment [15].  
Figure 1: Scanning process schematic.  
Based on the description above, several factors influence the acquisition failure probability. The light intensity reaching the acquisition camera affects its energy accumulation, the minimum threshold that can be taken affects the acquisition judgement, and the jitter associated with the scanning beam affects how effectively the scanning beam covers the UC along the scanning trajectory and affects the energy accumulation process. This section provides a detailed analysis of these factors. First, the main parameters relevant to the acquisition process for the Taiji program are presented in Table 1.  
During the acquisition process, the scanning beam moves along a pre-set curve centered on the UC. For the Taiji program, we use a constant tangential speed Archimedean spiral scanning pattern, which is the most efficient. A mathematical description of this scan trajectory is provided below [19]:  
\[\vec{r}=\frac{D_{t}}{2\pi}\phi(t)\left(\begin{array}{c}\cos\phi(t)\\ \sin\phi(t)\end{array}\right),\phi(t)=2\pi\frac{r_{max}}{D_{t}}\sqrt{\frac{t}{ T}}, \tag{1}\]  
where \(t\) is the time from the start of scanning to the current scan position, \(\vec{r}\) is the vector from the scan start point to the current position, \(T\) is the total scanning time, \(D_{t}\) is the track width between the two trajectories of the Archimedean spiral, and \(r_{max}\) is the maximum spiral radius, where \(r_{max}=\theta_{uc}+D_{t}/2\).  
Assuming a constant angular speed of \(v_{t}\), the total scanning time for the entire UC is provided by the following [16]:  
\[T=\frac{\pi r_{max}^{2}}{D_{t}v_{t}}. \tag{2}\]