## 5 Conclusions  
An accurate centroid estimation of beacon images is required to ensure a beacon tracking system will precisely point an optical beam toward a partner terminal, and provide a reliable optical communication link even in the presence of atmospheric turbulence. However, centroid accuracy is limited by finite delay time as well as noise sources including detector and background noises. Centroid error is also influenced by intensity fluctuations resulting from atmospheric turbulence, which can terminate the beacon tracking link.  
In this study, we analyzed the performance of two beacon tracking detectors onboard small satellites under atmospheric turbulence in terms of two metrics: centroid error and fade probability. Taking into account the log-normal tracking channels, the closed-form expressions were derived for not only the average centroid error, but also fade probability. We investigated two metrics for CMOS and QAPD detectors through numerical simulations by considering the windowed readout mode of the CMOS detector for high bandwidth, and confining finite delay time to the exposure time latency. As the exposure time became longer, the average centroid error due to noise sources decreased, but the centroid error caused by exposure time latency increased exponentially.  
Atmospheric turbulence degrades the performance of the beacon tracking system, which is a crucial factor in causing the large centroid error in the regimes of long exposure time rather than a short one (i.e., >1 ms) due to the angle of arrival fluctuations. In addition, it makes a big difference in fade probabilities between weak and moderate turbulence conditions, on 3 or 4 orders at the exposure time of 0.1 ms, and the longer the exposure time, the greater the difference.  
An optimal exposure time exists that produces the smallest total centroid error in the presence of atmospheric turbulence. It ranges from 0.2 ms to 0.5 ms, depending on the detector type and the strength of atmospheric turbulence, and where the total centroid error is to the order of micro radians. The QAPD detector showed better performance for total centroid error than the CMOS detector, while the CMOS detector provided superior performance in terms of the fade probability. Thus, two metrics have to be taken into account in the PAT system design to improve beacon tracking accuracy and to ensure reliable beacon tracking link.  
Figure 7: Fade probability versus exposure time under two atmospheric turbulence conditions.  
**Author Contributions:** Conceptualization, H.-C.L.; Formal analysis, H.-C.L. and C.-S.C.; Software, K.-P.S.; Writing--Original draft, H.-C.L.; Writing--Review & editing, C.-S.C. and J.-U.P.; Project administration, M.C. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** All data generated or analyzed during this study are included in this article.  
**Acknowledgments:** This work was supported by the Korea Astronomy and Space Science Institute through the project \"Operation of Observation Facilities for Space Object Surveillance\" funded by the Ministry of Science and ICT (MSIT) of the Korean government.  
**Conflicts of Interest:** The authors declare no conflict of interest.