## 3 Results
### Nadir Point Track and FOV
The time window is taken from one day to 27 days because the revisiting period of the Moon and L1 with respect to Earth is approximately one day and the orbit period of the halo orbit is always less than half of the lunar orbit. One year of data as a calculation period is sufficient to show the main patterns of the nadir point's latitude variation. Figure 4a-d revealed that in different years, which means different inclinations of the lunar orbit, the temporal complexity curves of latitude variation have similar shapes in both cases of the Moon and halo orbits, thus, they are independent of the inclination of the lunar orbit. On the one hand, considering the complexity patterns manifested by the shape of the curve, it is obvious that the shape of curves in the case of the Moon is an inerratic near sinusoidal curve which is determined by the approximate uniform circular motion of the Moon around the Earth, while the shape of curves in case of the halo orbit shows two peaks when the time window is around 8 days and 17 days. On the other hand, for each time window, the quartiles and medians of latitude variation in the case of a halo orbit are approximately equal to that of the Moon; while the maximum variation of latitude for each time window in the case of a halo orbit is greater than that of Moon especially around the peaks, which is around 14\({}^{\circ}\) and 21\({}^{\circ}\), respectively, for a 20,000-km halo orbit. The above demonstrates that the halo orbit induced complexity of the nadir point's latitude variation by splitting one peak into two while it can also extend the range of latitude variation for each time period. Figure 3: Nadir point tracks of the Moon (**a**) and a halo orbit (**b**) at 00:00:00 UTC each day from 20 March 2024 to 20 March 2025 when the lunar orbital inclination reaches the maximum of 28\({}^{\circ}\). As the name suggests, the halo orbits are like lunar aureole viewing from the perspective of the Earth due to the same angular momentum around the Earth, and the distance of the L1 to the Moon is much smaller than to the Earth, which leads to a similar field of view of the Earth. Figure 5 shows the variation of the field of view (FOV) of the Earth from halo orbits with z-directional amplitudes of 10,000 km (a) and 20,000 km (b) and the mean FOV (c) in a year from 20 March 2024 to 20 March 2025 varying with z-directional amplitude, respectively. It can be seen that in the general range, both have roughly the same FOV and only when the orbit is very large will there be a significant difference. Thus, there is not much difference from the perspective of observing the Earth as a whole. Figure 4: The temporal complexity of latitude variation at the nadir point of the 20,000-km halo orbit (**a,c**) and Moon (**b,d**). The horizontal axis represents the length of the sliding time window, and the vertical axis represents the range of latitude variation within the time window; each plotted box counts the latitude variations of the nadir point calculated using one-hour time steps over a one-year period from 20 March 2024 to 20 March 2025 (**a,b**) and from March 2033 to March 2034 (**c,d**).