## 3 Orbit Design
In addition, the same perigee argument has been considered, to minimize the instantaneous satellite velocity differences and in turn, the along-track relative displacements [17]. The remaining design parameters (\(\Omega\) and \(M\)) define the difference between the orbits of the two satellites. The differences between the ascending node right ascensions (\(\Delta\Omega\)) and the mean anomalies (\(\Delta M\)), _i.e._, the relative position of the satellites (shown in Figure 3), have been computed by means of the formulas provided in [17]. \(\Delta M\) has been chosen in order to enable the passive satellite to be located in the incidence plane of the illuminator at the initial time, thus fixing the observation geometry when the receiver is over the equator. Such a condition corresponds to \(\varphi_{\rm s}=180^{\circ}\), in this case. Figure 3 shows the position of the active and passive satellites when the latter is over the equator and the line joining the two positions (_i.e._, the baseline). This direction has to be comprised within the azimuth aperture of both the transmitting and receiving antennas. To calculate \(\Delta\Omega\) and \(\Delta M\), we have assumed \(\theta_{\rm s}\) equal to the maximum among those suggested in [2] (_i.e._, \(35^{\circ}\)), in order to increase the range of latitudes for bistatic acquisitions [17]. As for \(\theta_{\rm s}\) at the initial epoch, we have firstly considered that the receiver should be in the backward quadrant because of the constraint we have imposed for spatial resolution. Then we have made the hypothesis that, for a soil moisture application, the minimum bistatic swath should be 10 km. By choosing \(\theta_{\rm s}=0^{\circ}\) at the initial epoch, the time interval during which \(\theta_{\rm s}\) is in a useful range (_i.e._, [0\({}^{\circ}\)-8\({}^{\circ}\)]) when the orbits are propagated is maximized (note that at the equator the baseline is at its maximum, as will be discussed in Section 4). However, looking at Figure 4, that shows the observation geometry at the initial time, it can be noted that such a choice would imply a very small bistatic swath at the equator (theoretically, only one point), because for \(\theta_{\rm s}=0^{\circ}\) the receiver is just above the boundary of the area illuminated by Envisat, with the maximum allowable zenith angle (_i.e._,  
Figure 3: Relative position of the satellites at the initial epoch (\(\gamma\) is the yaw-steering angle of the active satellite). 35\({}^{\circ}\)), and cannot accomplish a forward observation. For the same reason, the ISMs whose far range looking angle is less than 35\({}^{\circ}\) (see Table 1) do not perform bistatic acquisitions over the equator, as will be shown in Section 4. The receiving near range point (NR\({}_{\rm RX}\)) has to be located within the area illuminated by the transmitting antenna (see Figure 4) and its distance to the transmitter far range point (FR\({}_{\rm TX}\)) cannot be less than 10 km. Hence, the best choice for the initial \(\theta_{\rm s}\) turned out to be 1\({}^{\circ}\), which is (approximately) the minimum zenith scattering angle for which the overlapping between the footprints of the antennas occurs with the selected minimum width of the bistatic swath. Table 2 reports the selected configuration and the orbital parameters at the initial epoch, that is the static design. The Satellite Tool Kit (available at: [[http://www.agi.com](http://www.agi.com)]([http://www.agi.com](http://www.agi.com))) has been successively used to propagate the orbits, taking into account only the orbital perturbations due to the \(J_{2}\) geo-potential harmonic, to maintain a heliosynchronous orbit and assuming negligible the orbital decay (_i.e._, the effect of the differential aerodynamic drag), as done in [18].