## 4 Discussion  
As Earth's sole natural satellite, the Moon and its several special Lagrange points in the cis-lunar space provide significant opportunities for lunar and deep space exploration, space weather and Earth observation, communication relay, and navigation service. Known scientific missions to Lagrange points have mainly focused on the Sun-Earth system while missions within the Earth-Moon system are relatively rare, but recently lunar exploration has led to an increase in the exploitation of Earth-moon Lagrange points especially L1 and L2. However, as for Earth observation, there have been limited studies on the L1 point of the Earth-Moon system in Earth observation in contrast to either Moon-based Earth observation or the L1 point of the Sun-Earth system. Recent investigations into Earth radiation monitoring at the L1 point have made the unrealistic assumption that a sensor could remain stationary at this location--an improbable scenario. Hence, this study endeavors to analyze the fundamental characteristics of the L1 point with respect to Earth observation. This study has three main improvements. The first, as mentioned above, is the determination of periodic halo orbits around the L1 point of the Earth-Moon system, which is designed for spacecraft to maintain its adjacent position in the neighborhood of the L1 point with lower energy consumption compared to the in situ maintenance. As for the calculation of halo orbit, it is still an idealized problem in the field of astronautics as there are many perturbation sources resulting in orbital errors such as the solar gravitational perturbation, thus it is necessary for spacecraft to carry out real-time orbit control. Therefore,  
Figure 8: Annual coverage cycle and frequency of eleven typical locations of different latitudes during the annual period from March 2024 to March 2025 (**a**) and from March 2033 to March 2034 (**b**). The grid of the time axis represents a sidereal month. the precise orbit determination in this paper is impossible as well as unnecessary. Anyway, this paper still considered the perturbation effect of main sources by adopting ephemeris data to accurately determine the position and relative motion of the Moon and Earth. Note that a common assumption in the N-body problem is that the mass of spacecraft can be considered to be zero relative to other main and perturbing celestial bodies. Thus this paper considers the perturbation from main sources to the motion of the Earth and Moon by means of ephemeris data while the direct gravitational perturbation to massless spacecraft is ignored, which is the second main improvement of this study. The final contribution of this study lies in the proposal of a \"temporal complexity curve\" describing the latitude variation at the nadir point, which effectively captures the intricate dynamics induced by halo motion, especially in contrast to the Moon's nadir point. This curve not only reveals patterns and complexities in latitude variation over specified time windows but also quantifies latitude variation values within those windows. Such quantification proves valuable for assessing ground coverage at different latitudes in the context of hemispheric-scale Earth observations. Leveraging the aforementioned enhancements, this study undertakes a comprehensive analysis of the general geometric characteristics of the L1 point in Earth observation, juxtaposed with Moon-based platforms for comparison. The results and subsequent analysis, focusing on nadir point variation, angular distribution, and spatial and temporal coverage in Section 3, affirm the strikingly similar Earth observation capabilities of L1 and Moon-based platforms. Both are adept at near-hemispheric-scale coverage of the Earth's surface within approximately one day and complete global coverage within a month. Notably, both achieve nearly daily revisits for mid-to-low latitudes--an unattainable feat for low Earth orbit and geostationary satellites.