# Parameter Design for Inter-Satellite Laser Link Acquisition Under Weak Light Conditions
## 7 Conclusions  
Inter-satellite link acquisition is a prerequisite for achieving GW detection missions. However, there is currently a lack of unified and in-depth research on situations where the optical power of the beam, after ultra-long distance transmission, becomes weak upon  
Figure 11: The probability of acquisition failure against changes in scan speed (**a**) and against changes in track width (**b**). The solid lines indicate the theoretical results, and the dashed lines indicate the simulation results.  
Figure 12: Mean acquisition time.  
reaching the receiving SC, as well as in scenarios where there are limitations on scan speed due to the use of a telescope-pointing system during the acquisition process. In this work, by analyzing the laser spot positioning error under weak light conditions applicable to space-based GW detection, we introduce a corresponding minimal photoelectrons threshold for acquisition judgment. Based on this, we derive the theoretical formula for acquisition failure probability by calculating the maximum acquisition radius. A simulation program is developed to verify the accuracy of the theoretical formula by closely mimicking the real acquisition process. The theoretical results are crucial for selecting optimal parameters by evaluating acquisition time. For the Taiji program, the key parameters are a scan speed of 3.31 \(\upmu\)rad/s, a track width of 0.87 \(\upmu\)rad, and an integration time of 800 ms, resulting in an acquisition time of 223.77 s for a single link.  
Through both simulation and theoretical results, it is evident that the acquisition failure probability varies significantly with scanning speed after introducing a photoelectron threshold for laser spot location under weak light conditions. This highlights the importance of carefully selecting the scanning speed when designing the acquisition system in the telescope-pointing scheme, serving as a key reference for the design of satellite attitude controllers during the acquisition process. Additionally, calculating the mean acquisition time helps estimate the time required to establish the laser link, although this calculation only accounts for the scanning time, excluding the time needed for satellite adjustments during acquisition. Overall, this work provides essential support for the design of intersatellite link acquisition parameters for the Taiji program, which can also be applied to other LISA-like space-based gravitational wave detection missions.  
Conceptualization, M.Z. and R.G.; methodology, M.Z. and R.G.; software, M.Z.; validation, M.Z., J.S., J.W. and P.L.; formal analysis, M.Z.; investigation, M.Z.; resources, P.L. and R.G.; data curation, M.Z., J.S. and J.W.; writing--original draft preparation, M.Z.; writing--review and editing, M.Z. and R.G.; visualization, M.Z.; supervision, R.G., P.L. and Z.L.; project administration, R.G.; funding acquisition, P.L. and Z.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Key Research and Development Program of China, grant number 2022 YFC2203702.  
Data are contained within the article.  
Thanks to the authors of this paper for their work on the research content. The authors thank the journal staff for their comments and revisions to this paper.  
The authors declare no conflicts of interest.