The number of satellites in the current low-Earth-orbit (LEO) satellite networks continues to grow to form a mega LEO satellite constellation (MLSC). This large-scale net-working effectively improves network coverage and capacity. However, denser satellite deployment brings more severe chal-lenges to satellite handovers, such as frequent handovers, which exponentially reduce the system performance (e.g. probability of service success (PSS)) when considering inherent handover failures. To ensure service continuity, this paper focuses on the relationship between the constellation scale and handover times under seamless coverage. Specifically, we first conduct spatial geometric analysis and probabilistic analysis to derive three new conditions for MLSC seamless coverage. Then, we analyze the tradeoff relationships between the handover times and satellite coverage duration with the given constellation scale. Furthermore, we construct a mathematical relationship between the constellation scale, handover times, and PSS, indicating the tradeoff between constellation scale and system performance. The analysis effectively guides to design or adjust the constellation scale, satellite altitude, satellite coverage angle, and handover strategy according to the requirements of system performance, which has important theoretical value for MLSC system design and future research.