## 2 Orbit Selection Approach
\[\left\{\lambda_{sorted,1},\ldots,\lambda_{sorted,k}\right\}=sort(\left\{ \lambda_{1},\ldots,\lambda_{k}\right\}) \tag{6}\]  
The largest and the smallest difference of the cyclic sequence of sorted longitudes are  
\[\Delta\lambda_{max,k}=\max\bigl{(}\left\{\lambda_{sorted,2}-\lambda_{sorted,1}, \ldots,\lambda_{sorted,k}-\lambda_{sorted,k-1},\lambda_{sorted,1}+360^{ \circ}-\lambda_{sorted,k}\right\}\bigr{)} \tag{7}\]  
and  
\[\Delta\lambda_{min,k}=\min\bigl{(}\left\{\lambda_{sorted,2}-\lambda_{sorted,1}, \ldots,\lambda_{sorted,k}-\lambda_{sort,k-1},\lambda_{sorted,1}+360^{ \circ}-\lambda_{sorted,k}\right\}\bigr{)}, \tag{8}\]  
respectively. Since we are interested in the denseness of the spatial sampling, we calculate these differences for \(k=1,\ldots,K\) orbital revolutions to find when the largest difference in longitude is notably reduced, i.e., when the spatial sampling densifies. Both differences are bounded by \(360^{\circ}/k\), which is the lower bound for \(\Delta\lambda_{max,k}\) and the upper bound for \(\Delta\lambda_{min,k}\). Figure 3 illustrates the differences and the bounding value of \(360^{\circ}/k\) for the first 90 orbital revolutions of the circular orbit presented in \[h_{l}=\frac{\Delta\lambda_{max,l}}{\Delta\lambda_{min,l}} \tag{1}\]0 The differences typically remain constant for a number of orbital revolutions before they are notably reduced, i.e., they depart from the bounding value of \(360^{\circ}/k\) and then make a step back towards that value. To find such steps, we check when the largest difference in longitude is reduced from one orbital revolution to the next one, more than its lower bound; i.e., we search for the index \(l\), for which  
\[h_{l}=\frac{\Delta\lambda_{max,l}}{\Delta\lambda_{min,l}} \tag{1}\]1  
\[h_{l}=\frac{\Delta\lambda_{max,l}}{\Delta\lambda_{min,l}} \tag{1}\]2 Largest and smallest differences in the longitudes of the ascending equator crossings of a circular orbit with a semimajor axis of 6,718,085 m and an inclination of \(70^{\circ}\). Black circles mark the largest differences that are notably smaller than those of the preceding orbital revolution. The black line represents the lower and upper limits for the largest and smallest differences, respectively, and represents exact homogeneity for each repeat period with the indicated number of orbital revolutions. The differences that fulfill Equation (9) are marked by black circles in \[h_{l}=\frac{\Delta\lambda_{max,l}}{\Delta\lambda_{min,l}} \tag{1}\]3 We calculate the ground track homogeneity only for these differences, which we signified by using the same index \(l\) in Equations (1) and (9). In this way, we identify at which numbers of orbital revolutions the ground track is near homogeneous, i.e., when we achieve a dense spatial sampling with a minimum number of orbital revolutions. Since the altitude is the parameter to optimize, we calculate the homogeneity for a large number of orbits whose orbital elements are identical except for the semimajor axis. In practice, we repeat the calculations for all values of the semimajor axis in the range from \(a_{1}\) to \(a_{2}\) in steps of \(\Delta a\). While the range is typically predefined, we need to select the step size \(\Delta a\). For that purpose, we determine how much the longitude of the last ascending equator crossing, \(\lambda_{K}\), changes when the semimajor axis is altered by \(\Delta a\). This change in longitude, denoted by \(\Delta\Delta\lambda_{K}\), can be derived from the differential of Equation (2)  
\[h_{l}=\frac{\Delta\lambda_{max,l}}{\Delta\lambda_{min,l}} \tag{1}\]4  
where  
\[h_{l}=\frac{\Delta\lambda_{max,l}}{\Delta\lambda_{min,l}} \tag{1}\]5  
and  
\[h_{l}=\frac{\Delta\lambda_{max,l}}{\Delta\lambda_{min,l}} \tag{1}\]6  
We select the step size \(\Delta a\) such that the longitude of the last ascending equator crossing changes at most by  
\[h_{l}=\frac{\Delta\lambda_{max,l}}{\Delta\lambda_{min,l}} \tag{1}\]7  
which is the smallest possible reduction of \(\Delta\lambda_{max,k}\) from the second-last to the last orbital revolution.