# Parameter Design for Inter-Satellite Laser Link Acquisition Under Weak Light Conditions
## 3 Analytical Model
### Acquisition Failure Probability Calculation
#### 3.1.2 Receiving SC in the Pre-Estimated UC  
In the second case, we assume that the receiving SC is located somewhere between track \(n\) and \(n+1\). Although the position of the receiving SC follows a Gaussian distribution within the UC, we can approximate it as being uniformly distributed between two tracks if the distance between the tracks is small compared to the entire radius of the UC. To ensure complete scanning of the entire UC and to account for beam jitter that deviates from the intended trajectory, the track width \(D_{t}\) introduction in Section 2 is less than the effective diameter of the scanning beam, as shown in Figure 5.  
The position of the satellite between track \(n\) and track \(n+1\) is denoted by the variable \(X_{t}\), while the jitter of the beam when it sweeps over the satellite is denoted by \(X_{n}\). The jitter is assumed to follow a Gaussian distribution with zero mean and a variance \(\sigma_{j}\), with a probability density function provided by a Gaussian function \(g(x_{n})\). Due to the presence of jitter, the receiving SC can be scanned not only by the left half of the beam along track \(n\) but also has a probability of acquisition by the right half of the beam. Therefore, the probability of the satellite not being successfully acquired along the track \(n\) scan can be expressed as [17]:  
\[X_{t}+X_{n}<-\theta_{div},\ \ X_{t}+X_{n}>\theta_{div}. \tag{14}\]  
Based on this, the probability of acquisition failure when the satellite is located at \(X_{t}=x_{t}\) can be calculated by the following:  
Figure 5: Schematic of the acquisition model, which shows that the receiving SC could carry out acquisition using the scanning beam on multiple trajectories. \[P_{n}(x_{t}) =P(X_{n}>\theta_{div}-x_{t})+P(X_{n}<-\theta_{div}-x_{t})\] \[=\int_{\theta_{div}-x_{t}}^{\infty}dx_{t}g(x_{n})+\int_{-\infty}^{- \theta_{div}-x_{t}}dx_{t}g(x_{n})\] \[=\frac{1}{2}erfc\left(\frac{\theta_{div}-x_{t}}{\sqrt{2}\sigma_{j }}\right)+\frac{1}{2}erfc\left(\frac{\theta_{div}+x_{t}}{\sqrt{2}\sigma_{j}} \right)\]  
Similarly, we can obtain the acquisition failure probability for the \(n+k\) trajectory by calculating the probability of acquisition failure using \(X_{t}+X_{n+k}<-\theta_{div}+k\cdot D_{t}\), \(X_{t}+X_{n+k}>\theta_{div}+k\cdot D_{t}\). By integrating the distribution of the satellites, we can obtain the acquisition failure probability:  
\[P_{tot}(x_{t})=\prod_{k=-N+1}^{N}\frac{1}{2}\Bigg{[}erfc\left(\frac{\theta_{div }+x_{t}-k\cdot D_{t}}{\sqrt{2}\sigma_{j}}\right)+erfc\left(\frac{\theta_{div} +k\cdot D_{t}-x_{t}}{\sqrt{2}\sigma_{j}}\right)\Bigg{]}, \tag{16}\]  
\[P_{tot,2}=\frac{2}{D_{t}}\int_{0}^{\frac{D_{t}}{2}}dx_{t}P_{tot}(x_{t}), \tag{17}\]  
where \(P_{tot}(x_{t})\) is the product of failure probabilities of the individual tracks for a given position \(x_{t}\), and \(P_{tot,2}\) is the total acquisition failure probability that acquisition was not completed on the 2N trajectories adjacent to track \(n\) in the second case. \(erfc(x)\) is the complementary error function. However, there are two prerequisites for this acquisition theoretical model: the acquisition camera's integration time is appropriate, and the power of the scanning beam is sufficient enough to enable acquisition within the effective radius \(\theta_{div}\). As discussed in Section 2.3, during the actual acquisition progress, the acquisition camera needs to accumulate enough energy. This energy accumulation is influenced not only by the position of the receiving spacecraft but also by parameters of the acquisition system, such as scan speed, track width, and the integration time of the acquisition camera. To facilitate the theoretical results, we assume that the horizontal axis value at which the scanning satellite vertically passes the receiving satellite and satisfies the minimum acquisition requirement within the integration time is defined as the maximum acquisition radius \(R_{m}\).