## 3 Application to the MAGIC Mission
### Five-Day Subcycle  
Table 1 lists near-optimal orbits for the 5-day subcycle sampling for the inclined and polar pair. In order to create a good constellation encompassing the inclined and polar pair, it is necessary as a first approximate step to choose between the different listed cases, selecting the combinations with minimal difference in longitude shift. When doing so, for each targeted subcycle, the two pairs could have an overlapping ground track crossing at the equator, or, probably better, interleaved tracks filling in gaps from the first pair by the second pair.  
Figure 6 illustrates an example of the ground tracks after 5 days for one of the optimized scenarios listed above. In particular, this figure shows the accumulated ascending ground tracks in 5 days for an inclined pair at 396 km and 65\({}^{\circ}\) of inclination and a polar pair at 434 km and 89\({}^{\circ}\). In this example, the orbits are chosen in order to have an overlap  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{1}{c}{**Satellite**} & \multicolumn{4}{c}{**5-Day Subcycle**} \\ \multicolumn{1}{c}{**Pairs**} & **Altitude (km)** & **Time Period (days)** & \(h_{I}\)**(c)** & \(\Delta\Delta_{shift}\)**(deg)** \\ \hline IP: **65\({}^{\circ}\)** & 388 & 4.871 & 1.403 & 1.622 \\  & 396 & 4.879 & 1.397 & \(-\)1.499 \\ \hline  & 393 & 4.877 & 1.144 & 0.636 \\ IP: **67\({}^{\circ}\)** & 453 & 4.877 & 1.203 & 0.827 \\  & 457 & 4.881 & 1.157 & \(-\)0.722 \\ \hline  & 397 & 4.882 & 1.168 & 0.736 \\ IP: **70\({}^{\circ}\)** & 457 & 4.882 & 1.218 & 0.880 \\  & 461 & 4.886 & 1.146 & \(-\)0.673 \\ \hline  & 352 & 4.898 & 1.417 & \(-\)1.652 \\ IP: **75\({}^{\circ}\)** & 405 & 4.891 & 1.157 & 0.692 \\  & 465 & 4.891 & 1.185 & 0.762 \\  & 469 & 4.896 & 1.173 & \(-\)0.794 \\ \hline  & 370 & 4.918 & 1.148 & \(-\)0.647 \\ PP: **87\({}^{\circ}\)** & 423 & 4.912 & 1.375 & 1.523 \\  & 425 & 4.914 & 1.167 & 0.733 \\ \hline  & 372 & 4.921 & 1.171 & \(-\)0.741 \\ PP: **88\({}^{\circ}\)** & 374 & 4.923 & 1.384 & \(-\)1.537 \\  & 490 & 4.919 & 1.145 & \(-\)0.669 \\ \hline  & 368 & 4.916 & 1.426 & 1.566 \\  & 376 & 4.925 & 1.411 & \(-\)1.631 \\ PP: **89\({}^{\circ}\)** & 434 & 4.924 & 1.383 & \(-\)1.458 \\  & 488 & 4.917 & 1.190 & 0.781 \\  & 492 & 4.921 & 1.172 & \(-\)0.790 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Orbit options for inclined pairs (IP) and polar pairs (PP) optimized as a first step for a subcycle of 5 days and a monthly subcycle between 28 and 32 days.  
of the ground track position of the satellites (inclined and polar pairs) at the ascending node (on the left in Figure 6) or to have interleaved positions at the equator to enhance the spatial sampling (on the right in Figure 6). Ground track near homogeneity for the two altitudes and inclinations is shown in Figure 7.  
Figure 6: Ascending ground tracks after 5 days for an inclined pair at 396 km and 65\({}^{\circ}\) of inclination (blue) and a polar pair at 434 and 89\({}^{\circ}\) of inclination (grey). This coincides with the so called scenario 5d_Ma. Two subsequent passages are highlighted with full (most recent) and shaded (past) color markers at the equator. **Left**: both pairs have the same equator crossings in the top global view and zoom in at the bottom. **Right**: both pairs have the interleaved equator crossings in the top global view and zoom in at the bottom.  
Figure 7: Ground track near homogeneity for the two altitudes and inclinations of the previous.