## 3 Numerical Experiments with Real Debris Orbital Data
In any case, the authors show that it is also possible to use the solar sail to reenter the GEO debris, but it will be necessary to consider an extremely large solar sail such that the SRP can increase the eccentricity so that the debris approaches the reentry zone. In the case of debris removal to the graveyard orbit, a value of \(A/m=1\) m\({}^{2}/\)kg is required, which is technically feasible with the existing technology. In [39], the sizes and shapes of the solar sails are discussed according to the available technology. Case 3: Now, we present the results for the case where the space debris will be directed to the reentry region, that is, close to Earth's atmosphere, to be incinerated. Since the \"stuffin.space\" site does not provide the data for the perigee argument \(g\) and the longitude of the ascending node \(h\) of the debris, we show diagrams of \(g\) versus \(h\) for a range of variation from \(0^{\circ}\) to \(360^{\circ}\) (see figure legends) to see if these orbital elements can contribute to an increase in the eccentricity. Figure 5a,b show, through dynamical maps, the amplitude of the eccentricity represented in the vertical color bar. Reference [27] shows the values of the solar sail's relation \(A/m\) such that the debris approaches the Earth's atmosphere, increasing its eccentricity due to the SRP acting on the sail. Thus, we choose the parameter \(A/m\) based on this work, namely, \(A/m=25\) m\({}^{2}/\)kg. Figure 5a shows that an object that presents a quasi-zero eccentricity (0.002) is strongly disturbed when taking into account a solar sail coupled to the object. Note that the amplitude of the eccentricity grows considerably until the object reenters the Earth's atmosphere, and this goal is achieved in the red regions. However, we observe that, for certain initial values of the longitude of the ascending node, and by means of the variations of the argument of perigee from zero to 360 degrees, we obtain constant ranges. However, we notice that, when we change the eccentricity of the object to 0.05, Figure 5b shows that these bands are destroyed and more complex regions appear in the dynamics. This means that, for objects in circular or near-circular orbits, the effects of the disturbing forces are even more accentuated. These results present a significant importance for the case of debris in the GEO region, as many of these bodies have almost zero eccentricities. Another point observed is that the presence of perturbations destroys completely the symmetry of the orbits. It is worth mentioning that in the most realistic case, where the Earth's atmosphere is present, when the object is excited, especially by the SRP, the eccentricity is strongly disturbed, so the SRP lowers the perigee of the debris orbit and consequently, with the  
friction of atmospheric drag, the debris does not return to the same apogee. In this way, it reduces the altitude of the orbit until the debris is incinerated. The characteristic of the orbital evolution of the debris is similar to the one appearing in Figure 5b. From these figures, it is possible to identify the values in which the orbit of the debris is more unstable. Namely, it means that, by directing the debris to values of the argument of perigee and longitude of the ascending node chosen from the maps, we can contribute to a decrease in the time of the orbital decay of the debris. Thus, we can save on reentry time and sail cost by choosing more appropriately the mass and area of the solar sail. These aspects contribute to the sustainability of space exploration, as they provide possibilities for removing space debris while using a clean and abundant propulsion mechanism that uses the energy of the Sun. Now, considering Figure 6a, we show the simulation data taking into account the TITAN 3C TRANSTAGE DEB 969-013D debris. Note that, for different values of \((g,h)\), we have different eccentricity amplitudes.