# Low-Thrust Transfer to Quasi-Synchronous Martian Elliptic Orbit via Nonlinear Feedback Control
## 2 Orbit Dynamics
### Operational Orbit and Coverage Analysis
The ground track associated with this orbit is portrayed in Figure 2, under the assumption of selecting RAAN and argument of latitude corresponding to zero geographical longitude and maximum latitude at the initial time. Two satellites are assumed to describe the same ground track. The second vehicle is delayed by \(N_{t}T_{n}/2\), i.e., half of the repetition period. Coverage analysis of this simple configuration aims at identifying (a) the minimum elevation for visibility, \(\varepsilon_{min}\), for those longitudes where continuous coverage is ensured, and (b) the maximum revisit time, \(t_{rev}\), for the remaining longitudes associated with discontinuous coverage. Both tasks (a) and (b) require obtaining the scheduling of the visible passes of the two satellites. However, due to the symmetry properties of the ground track, only a limited number of sample locations are to be considered to supply precise information on the global coverage properties. These sample locations are portrayed in Figure 3 (denoted with \"x\" and \"o\") and lie along the symmetry meridians of the ground track. In fact, let \(\varepsilon_{min}\big{(}\lambda_{g},\phi\big{)}\) and \(t_{rev}\big{(}\lambda_{g},\phi\big{)}\) denote, respectively, the minimum elevation angle (task (a)) and the maximum revisit time (task (b)) over a repetition period, as functions of latitude, \(\phi\), and geographical longitude, \(\lambda_{g}\). If \(\phi\) is set to a specific value \(\overline{\phi}\), then \(\varepsilon_{min}\big{(}\lambda_{g},\overline{\phi}\big{)}\) (and \(t_{rev}\big{(}\lambda_{g},\phi\big{)}\)) becomes a function of \(\lambda_{g}\), and is continuous and locally even with respect to 18 values of \(\lambda_{g}\), i.e., \(\lambda_{g}=20k\) deg \((\dot{k}=-9,-8,\ldots,8)\). These values correspond to the symmetry meridians of  
Figure 1: Quasi-synchronous orbits: \(q\) as a function of the eccentricity. Figure 2: Ground track of the operational quasi-synchronous, elliptic orbit. the ground track. This circumstance implies that the function \(\varepsilon_{min}\big{(}\lambda_{g},\overline{\phi}\big{)}\) (and \(t_{rev}\big{(}\lambda_{g},\phi\big{)}\)) has extremal values at these values of \(\lambda_{g}\). As a result, for a given latitude \(\overline{\phi}\), the minimum and maximum values of \(\varepsilon_{min}\big{(}\lambda_{g},\overline{\phi}\big{)}\) (and \(t_{rev}\big{(}\lambda_{g},\phi\big{)}\)) can be identified by considering only the locations marked with \"o\" and \"x\" in Figure 2.  
Visibility depends on the minimum elevation angle \(\varepsilon_{min}\). In this study, two distinct values of \(\varepsilon_{min}\) are considered: (i) \(\varepsilon_{min}=0\) deg and (ii) \(\varepsilon_{min}=5\) deg. Continuous coverage is guaranteed for all locations with either \(\phi\geq 50\) deg (in case (i)) or \(\phi\geq 60\) deg (in case (ii)). For each latitude, two distinct longitudes are considered, i.e., those marked with \"x\" and \"o\" in Figure 2, and the maximum and minimum value of \(\varepsilon_{min}\) over a repetition period are reported in \[a=\left\{\frac{\cos i}{\omega_{M}\Big{[}1-(\sin\omega\sin i)^{2}\Big{]}}\sqrt{ \frac{\mu(1-e)}{(1+e)^{3}}}\right\}^{\frac{2}{3}} \tag{15}\]0 The remaining latitudinal region is not covered uninterruptedly, and the maximum revisit times are identified and reported in \[a=\left\{\frac{\cos i}{\omega_{M}\Big{[}1-(\sin\omega\sin i)^{2}\Big{]}}\sqrt{ \frac{\mu(1-e)}{(1+e)^{3}}}\right\}^{\frac{2}{3}} \tag{15}\]1 For \(\phi=50\) deg only the revisit time when \(\varepsilon_{min}=5\) deg (case (ii)) is meaningful. Inspection of Tables 1 and 2 points out that continuous coverage can be guaranteed by only two satellites in a relatively large latitudinal region, while the maximum revisit time remains sufficiently short at mid-latitudes (30 and 40 deg). Moreover, at high latitudes, the minimum elevation angle has only modest variations depending on geographical longitude. This property is apparent from inspection of Table 1, where the extremal values of \(\varepsilon_{min}\) are reported.