## 4 Numerical Study
### Formation in Medium-Inclination Circular LEO  
In this section, the stability of a satellite formation designed to perform ELINT is investigated. The orbit parameters of the chief satellite and the formation configuration are reported in Tables 4 and 5 and result in the relative trajectory shown in Figure 5. \begin{table}
\begin{tabular}{c c c} \hline \hline semimajor axis & \(a\) & 6919 km \\ eccentricity & \(e\) & 0 \\ inclination & \(i\) & 52 deg \\ RAAN & \(\Omega\) & 0 deg \\ epoch (UTC) & - & 24 January 2023 12:00:00 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mid-inclination LEO formation: orbit parameters of chief satellite. Figure 1: SSO formation: \(J_{n}\)-invariant bounded trajectory (black) in \(\mathcal{F}_{r}\); the first orbit is plotted in red. Figure 2: SSO formation: trajectory (black) in \(\mathcal{F}_{r}\) resulting from non-invariant initial conditions; the first orbit is plotted in red. As for the case in Section 4.1, the variations in the extreme values of the position coordinates for the first and the last orbit, collected in Table 6, are compared. The results indicate that both the in-plane and the out-of-plane oscillations decrease by a marginal 1.2% and 2.7%. The use of initial conditions determined based on the unperturbed model results in the trajectory represented in Figure 6, characterized by a high drift rate along the \(\hat{y}\) direction and rapid increase in the amplitude of the oscillations, which can be observed in detail in Figures 7 and 8. Figure 4: SSO formation: time behavior of \(z\) component. Figure 3: SSO formation: time behavior of \(y\) component. \begin{table}
\begin{tabular}{c c c c} \hline \multicolumn{4}{c}{**Hamiltonian Variables**} & **Position/Velocity Components** \\ \hline \(w_{1}\) & \(0\) & \(x\) & \(9.7849\) km \\ \(w_{2}\) & \((1+\mathrm{j})\cdot 10^{-3}\) & \(y\) & \(-19.5697\) km \\ \(w_{3}\) & \((1+\mathrm{j})\cdot 10^{-3}\) & \(z\) & \(-9.7849\) km \\ \(z_{1}\) & \(0\) & \(v_{x}\) & \(-7.5402\) m/s \\ \(z_{2}\) & \((-1-\mathrm{j})\cdot 10^{-3}\) & \(v_{y}\) & \(-21.5257\) m/s \\ \(z_{3}\) & \((-1-\mathrm{j})\cdot 10^{-3}\) & \(v_{z}\) & \(0.2797\) m/s \\ \hline \end{tabular}
\end{table} 
Table 5: Mid-inclination LEO formation: initial state of deputy satellite. To further evaluate the effectiveness of the method developed in this research, relative trajectories computed from \(J_{n}\)-invariant initial states can be compared to the ones obtained integrating \(J_{2}\)-invariant initial states derived from the method proposed by [PERSON] and [PERSON], which is known to be effective especially for non-polar orbits. The orbit parameters of the chief satellite and the \(J_{2}\)-invariant initial state of the deputy satellite are the ones indicated in [15] and reported in Tables 7 and 8. The \(J_{n}\)-invariant initial state is reported in Table 9. It shall be noted that the model proposed here is developed for satellites operating along orbits with null or negligible eccentricity (see Appendix A.1); therefore, its application on an elliptic orbit, like the one in Table 7, shall be carefully evaluated. Because the error introduced in Equations (A1)-(A5) \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **First Orbit** & \multicolumn{2}{c}{**Last Orbit**} \\ \hline \(x^{max}\) & \(11.9577\) km & \(x^{max}\) & \(11.9611\) km \\ \(x^{min}\) & \(-11.9724\) km & \(x^{min}\) & \(-11.9790\) km \\ \hline \(y^{max}\) & \(18.0742\) km & \(y^{max}\) & \(19.2724\) km \\ \(y^{min}\) & \(-29.8000\) km & \(y^{min}\) & \(-28.5862\) km \\ \hline \(z^{max}\) & \(9.5223\) km & \(z^{max}\) & \(9.5223\) km \\ \(z^{min}\) & \(-9.4888\) km & \(z^{min}\) & \(-9.4888\) km \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Mid-inclination LEO formation: maximum and minimum value of the position coordinates for the first and the last orbit. Figure 5: Mid-inclination LEO formation: \(J_{n}\)-invariant bounded trajectory (black) in \(\mathcal{F}_{r}\); the first orbit is plotted in red.