## 3 Simulations
### Analysis of Results  
This subsection reports the interpretation of the results in Section 3.2. The first set of functions were the ones in (52). From Figure 2a-c, the dependence of \(\Delta\Omega\) with respect to the right ascension of the ascending node of the first satellite \(\Omega_{1}\) revealed that it was always decreasing, with a possible presence of a jump discontinuity in the value of \(\Omega_{1}=0\). Furthermore, \(\Delta\Omega\) appeared also to be anti-symmetric with respect to \(\Omega_{1}\), i.e., \(\Delta\Omega(\Omega_{1})=-\Delta\Omega(-\Omega_{1})\). This behavior confirms the prediction of a previous investigation [26], even if it was restricted to a selected instance of the problem. A possible approximate and linear model for \(\Delta\Omega\) with respect to \(\Omega_{1}\) to capture the behaviors of these results is  
\[\begin{split}&\Delta\Omega\Big{(}\Omega_{1},\
u_{1}^{\Lambda}, \Lambda_{*},r_{*}\Big{)}\approx-k_{\
u_  
In Figure 2c, it can be noticed also that the function \(\Delta\Omega\) with respect to the orbit radius \(r\) presented an amplification and a change of scale with respect to \(\Omega_{1}\), so  
\[\Delta\Omega\Big{(}\Omega_{1},\
u_{1,*}^{\Lambda},\Lambda_{*},r\Big{)}\approx-k_ {1}(r)\Delta\Omega\Big{(}-k_{2}(r)\Omega_{1},\
u_{1,*}^{\Lambda},\Lambda_{*} \Big{)} \tag{58}\]  
could also be an alternative possible model with \(k_{1}(r),k_{2}(r)\in\mathbb{R}\setminus\{0\}\). From Figure 2a,d,e, the dependence of \(\Delta\Omega\) with respect to \(\
u_{1}^{\Lambda}\) appears more complicated. In selected cases of [26], a symmetric behavior such as \(\Delta\Omega\big{(}\
u_{1}^{\Lambda}\big{)}=\Delta\Omega\big{(}-\
u_{1}^{ \Lambda}\big{)}\) was predicted, but the extended analysis of the present study reveals that \(\Delta\Omega\big{(}\
u_{1}^{\Lambda}\big{)}\) is still symmetric when varying the orbit radius \(r\) (i.e., \(h_{\text{S}}\)) but has more complicated behavior with respect to the latitude of the emitter when \(\Lambda\
eq 0\), revealing a local minimum and a local maximum, as shown in Figure 2d. As shown in Figure 2f, there was a decreasing behavior with respect to the orbit radius \(r\) (i.e., the height \(h_{\text{S}}\)). This is also evident in Figure 2e and for positive values of \(\Omega_{1}\) in Figure 2c, so  
\[ \begin{cases}\dfrac{\partial(\Delta\Omega)}{\partial r}>0&\Omega_{1}>0\\ \dfrac{\partial(\Delta\Omega)}{\partial r}<0&\Omega_{1}<0\end{cases}  \tag{59}\]  
is a possible interpretation of the data. The second set of functions were the ones reported in (53). From Figure 3a-c, the dependence of \(\Delta\
u\) with respect to \(\Omega_{1}\) reveals an anti-symmetric behavior, i.e., \(\Delta\
u(\Omega_{1})=-\Delta\
u(-\Omega_{1})\). In addition to anti-symmetry with respect to \(\Omega_{1}\), there was a translation with respect to \(\
u_{1}^{\Lambda}\), i.e.,  
\[\Delta\
u\Big{(}\Omega_{1},\
u_{1}^{\Lambda},\Lambda_{*},r_{*}\Big{)}\approx- \Delta\
u\Big{(}-\Omega_{1},\
u_{1}^{\Lambda},\Lambda_{*},r_{*}\Big{)}+k \Big{(}\
u_{1}^{\Lambda}\Big{)}\text{sgn}\Big{(}\
u_{1}^{\Lambda}\Big{)} \tag{60}\]  
with \(k\big{(}\
u_{1}^{\Lambda}\big{)}\in\mathbb{R}_{>0}\); the sign function modeled the mathematical jump. There was a translation also with respect to \(\Lambda\), i.e.,  
\[\Delta\
u\Big{(}\Omega_{1},\
u_{1,*}^{\Lambda},\Lambda,r_{*}\Big{)}\approx- \Delta\
u\Big{(}-\Omega_{1},\
u_{1,*}^{\Lambda},\Lambda,r_{*}\Big{)}+k( \Lambda)\text{sgn}(\Lambda) \tag{61}\]  
with \(k(\Lambda)\in\mathbb{R}_{>0}\). With respect to the orbit radius \(r\), the function \(\Delta\
u\) presented an amplification and a change of scale with respect to \(\
u_{1}^{\Lambda}\), so  
\[\Delta\
u\Big{(}\Omega_{1},\
u_{1,*}^{\Lambda},\Lambda_{*},r\Big{)}\approx-k _{1}(r)\Delta\
u\Big{(}-k_{2}(r)\Omega_{1},\
u_{1,*}^{\Lambda},\Lambda_{*} \Big{)} \tag{62}\]  
with \(k_{1}(r),k_{2}(r)\in\mathbb{R}\setminus\{0\}\) is a possible model to interpret the data in Figure 3c.