## 3 Application to the MAGIC Mission
### Recommended Orbits  
Based on previous Tables 1 and 2, it is possible to identify specific sets of orbits that are nearly nullifying the difference in the ground track longitude shifts for certain subweekly subcycles (Table 3). The inclinations and/or altitudes in Table 3 can be fine-tuned to make sure that the longitude shifts of the two pairs have an exact match. For example, in the worst case of scenario 3d_M, it would require a + 0.2\({}^{\circ}\) change in inclination or a 321 m change in altitude for the inclined pair. See Appendix B for the full table of matching combinations. The 5- and 7-day subcycle scenarios are identified by \"5d\" and \"7d\", respectively. The scenario ID is supplemented by \"M\" for medium and \"H\" for higher altitudes. These IDs are associated with having one or two sets at altitudes approximately between 400 and 450 km and over 450 km, respectively. Altitudes over 500 km are not recommended due to the too low sensitivity to the gravity signal and to the limit in achieving the user needs as shown in the Appendix B of the MAGIC MRD [36]. For completeness, two scenarios optimized for 3 days are also introduced in the following table to provide additional options, should higher priority be given to short time scales for example for near-real-time applications. As additional information, the last column of Table 3 provides a list of all the common subcycles achievable by both pairs. As previously mentioned, all the recommended scenarios include at least one subcycle between 28 and 32 days. As discussed before, higher importance is given to the homogeneity at short time scales. A more detailed version of Table 3 is provided in Appendix A. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Satellite**} & \multicolumn{4}{c}{**7-Day Subcycle**} \\ \cline{2-5}  **Pairs** & **Altitude (km)** & **Time Period (days)** & \(\mathbf{h_{1}}\)**(-)** & \(\mathbf{\Delta\lambda_{shift}}\)**(deg)** \\ \hline IP: \(\mathbf{65}^{\circ}\) & 428 & 6.854 & 1.258 & \(-0.757\) \\ \hline IP: \(\mathbf{67}^{\circ}\) & 474 & 6.859 & 1.219 & \(-0.723\) \\ \hline IP: \(\mathbf{70}^{\circ}\) & 389 & 6.861 & 1.238 & \(0.743\) \\  & 432 & 6.862 & 1.218 & \(0.670\) \\  & 478 & 6.866 & 1.203 & \(-0.669\) \\ \hline IP: \(\mathbf{75}^{\circ}\) & 397 & 6.875 & 1.222 & \(0.696\) \\  & 486 & 6.879 & 1.266 & \(-0.870\) \\ \hline PP: \(\mathbf{87}^{\circ}\) & 378 & 6.911 & 1.243 & \(-0.751\) \\  & 417 & 6.906 & 1.253 & \(0.786\) \\  & 462 & 6.910 & 1.215 & \(-0.643\) \\ \hline  & 419 & 6.909 & 1.204 & \(0.642\) \\ PP: \(\mathbf{88}^{\circ}\) & 461 & 6.908 & 1.284 & \(0.851\) \\  & 464 & 6.913 & 1.277 & \(-0.804\) \\ \hline \hline PP: \(\mathbf{89}^{\circ}\) & 463 & 6.911 & 1.226 & \(0.692\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Orbits for inclined pairs (IP) and polar pairs (PP) optimized for a subcycle of 7 days and a monthly subcycle between 28 and 32 days. Figure 8 illustrates the sensitivity to altitude changes (\(\pm\)2 km) around the altitudes of the 5d_Ma orbit candidate, while Table 4 quantifies the variations for \(h_{l}\) and longitude shifts for all the recommended orbits. The change in homogeneity and longitude shift of the 3-day subcycles can be around 2 and 5 times less sensitive to a change in altitude than the 5- and 7-day examples, respectively.