## 3 Solar Coronagraph Trajectory Design  
In low-Earth orbit, the Sun is an inertial vector during a couple of orbital periods, so it shows a constant cross-track component. Therefore, the out-of-plane and in-plane motions of the Sun must be distinguished, and the less out-of-plane motion, the more efficient the staging mode. Assuming a sun-synchronous orbit, the changeable orbital elements are the semi-major axis and RAAN (\(\Omega\), Right Ascension of the Ascending Node). The altitude was set as 550 km, so the influence of the RAAN should be determined. The RAAN, with respect to the Sun, can be expressed as the LTAN. Therefore, regardless of the position of the Sun, the closer the LTAN is to 00:00 or 12:00, the better the alignment. The position of the Sun in the Earth-centered inertial frame (ECI frame) is expressed as an ecliptic plane. The vernal equinox, the intersections of the ecliptic plane, and the Earth's equator plane define the x-axis of the ECI frame (Figure 4a). Because the maximum declination of the Sun for one year is 23.6\({}^{\circ}\), the z-component of the Sun is always smaller than the magnitude of the vector projected onto the XY plane. The launch date of the CANYVAL-C mission was planned to be March, so the position of the Sun was set at the vernal equinox. Figure 4b shows the geometrical relationship of the sun-synchronous orbit according to the LTAN and Sun vector. If the LTAN differs by 12 h, the ascending and descending nodes appear on opposite sides of the same plane. According to the definition of the LTAN, the position of the ascending node with respect to the Sun (\(\delta\Omega\)) differs by 15\({}^{\circ}\) for every hour of the LTAN. Because the inclination is not exactly 90\({}^{\circ}\) for the actual sun-synchronous orbit, the angle between the orbital plane and the Sun (\(\beta\)) is calculated and presented in Table 2. An orbit with a large along-track component was the case for LTAN at 01, 11, 13, and 23. This indicates a difference of less than 3.5\({}^{\circ}\) between \(\delta\Omega\) and \(\beta\). \(\sin\beta\) is the cross-track ratio of the Sun vector, ranging from 0.23 to 0.97. The reference trajectory considering the position of the Sun in the LVLH frame is presented in Figure 5. From the left panel of Figure 5, the closer the LTAN is to 00 or 12, the smaller the cross-track component, and the closer it is to 06 and 18, the larger the cross-track component. According to the right panel in Figure 5, the cross-track component is constant, and the in-plane components change continuously. Table 3 shows the daylight duration (\(T_{day}\)) and ratio of the daylight duration to the orbital period (\(T_{day}/T\)). The orbital period was 95.64 min for every orbit, and the daylight period was approximately 63% for the LTAN 01, 11, 13, and 23 orbits and 100% for the LTAN 05, 07, 17, and 19 orbits. If the cross-track component of the LTAN is large, it shows a dawn-dusk orbit with long exposure to the Sun. As a result, there is no eclipse, which suggests many opportunities for alignment with the Sun. On the other hand, when the cross-track component is small, alignment in the along-track direction is easy, but the daylight duration is shortened, resulting in fewer opportunities for alignment. Figure 3: Conceptual description for IAH requirement.