## 3 Application to the MAGIC Mission  
The approach described in the previous section was used to identify a starting set of orbits for the Phase A studies of the NGGM mission concept. Achieving accurate and purely satellite-based solutions on daily to weekly time scales is not possible with the current generation of gravity missions. MAGIC aims at responding to this objective to improve current models and in particular to introduce the capability to support monitoring and mitigating extreme events. For this reason, particular attention was paid to weekly/subweekly subcycles in the MAGIC orbit candidates' selection process. Another benefit of the choice of these near-homogeneous short periods is that, for example, weekly variations within each month are separable from the monthly variations, thus resulting in improved monthly solutions.  
The so-called Bender constellation consisting of two satellite pairs flying at two different orbit inclinations was identified as the most promising concept to meet the global user community requirements and is currently the baseline for MAGIC. This constellation type is favored over other formations due to its good performance for the gravity field retrieval and its implementation options allowing different spatial resolution results to be obtained for different periods. As stated in version 1 of the Mission Requirements Document (MRD) [36], the constellation shall consist of two pairs: the first one in a near-polar orbit and the second one with an inclination between 65 and 70\({}^{\circ}\). A deviation from these inclinations, such as up to 75\({}^{\circ}\), can be investigated if another orbit sampling/coverage is favorable for specific applications. In addition, the mission shall provide a near-homogeneous sampling over a subcycle of 3-7 days. To quantify the density of the spatial sampling at a predefined number of orbital revolutions, we can use the ground track homogeneity parameter (\(h_{l}\)). In order to select the optimal orbits for the two pairs, it is necessary to search for the altitudes that offer small homogeneity values for a number of periods common to both, which relate to the desired temporal resolutions and fulfill the constraint of the same ground track shift for both pairs at the targeted subweekly subcycle. Thus, the combination of the homogeneity and ground track shift in longitude and the interleaving of the sampling of both pairs at the equator enables optimized sampling of the individual pairs as well as the constellation.  
In this section, we discuss the orbit candidates for a polar pair (PP) and an inclined pair (IP), which we have identified using the method introduced in Section 2. The orbits are assumed to be at a constant mean altitude over the mission lifetime and the solutions are optimized for 5- and 7-day subcycles. Beyond this subweekly sampling, the presented scenarios offer at least another subcycle between 28 and 32 days, which guarantees for monthly solutions a near-homogeneous ground track as well. The ground track homogeneity is considered to range between 1.0 and 1.5 as a goal and 1.5 and 2.0 as a threshold. Since shorter periods have a lower ground track density than longer periods for the same value of \(h_{I}\), the subweekly samplings are selected with a more rigid constraint of \(h_{I}<1.5\).  
A large number of inclinations are introduced to illustrate the sampling capabilities. Inclined pairs are studied for inclinations of 65, 67, 70 and 75\({}^{\circ}\), while polar pairs are investigated for 87, 88 and 89\({}^{\circ}\). For the polar pair, a high inclination is preferred to avoid polar gaps. However, in this initial phase, lower inclinations (e.g., 87 and 88\({}^{\circ}\)) were also introduced to provide a broader range of sampling options for the two pairs.