## 3 Numerical Experiments with Real Debris Orbital Data
In two specific regions (dark color), the debris obtains the required eccentricity value to reach the reentry region (see the eccentricity values on the scale of colors). The blue region shows the values of \((g,h)\) that keep the orbit relatively high in eccentricity, but do not allow the debris to reach the reentry region. Figure 6a shows the conditions of \((g,h)\) that the debris must be directed to contribute to the amplification of the eccentricity and thus reenter the Earth's atmosphere when atmospheric drag is taken into account in the dynamics. Figure 6b shows the variation in the same angles shown in Figure 6a, but now the color scale shows the time when the debris reaches the reentry region. Figure 6b shows the time when the space debris reaches the reentry zone (radius of Earth + 120 km) for the values of \((g,h)\), which are the reddest regions in Figure 6a. Note that only a few values of \((g,h)\) contribute to amplifying the eccentricity, so that the debris can be directed to the reentry zone with a value of \(A/m=25\) m\({}^{2}\)/kg. As shown in the figure, for some values of \((g,h)\) the debris reenters very quickly, a little over a year, while in a small band (in yellow) it takes a little more than three years for that. This analysis is very important to identify possible regions of instability where the eccentricity of the debris can grow very quickly, causing the debris to reach the reentry region at the perigee of its orbit. Depending on the value of \((g,h)\), the reentry time changes, as shown in the color scale of Figure 6b. In this case, the value of the ratio \(A/m=25\) m\({}^{2}\)/kg, which in terms of technical feasibility is not possible due to the size required for the sail. The ESA, NASA, and other space agencies are working to produce very large solar sails. Performing a more refined search, we can obtain suitable regions for the values of \((g,h)\) in which a lower \(A/m\) ratio can be found and that can amplify the eccentricity variations. For this, a larger number of simulations must be performed. In any case, this value will be considered high in the case of debris in GEO orbit, which, to be pushed into the region of the Earth's atmosphere, needs a large solar sail. It is worth noting that the ESA ([[https://wwwesa.int/Enabling_Support/Space_Engineering_Technology/Shaping_the_Future/Sail_solutions_for_space_junk](https://wwwesa.int/Enabling_Support/Space_Engineering_Technology/Shaping_the_Future/Sail_solutions_for_space_junk)]([https://wwwesa.int/Enabling_Support/Space_Engineering_Technology/Shaping_the_Future/Sail_solutions_for_space_junk](https://wwwesa.int/Enabling_Support/Space_Engineering_Technology/Shaping_the_Future/Sail_solutions_for_space_junk)), accessed on 7 September 2022) is building a prototype solar sail with an area of 25 m\({}^{2}\), which would be suitable for a spacecraft weighing around 700 kg. Moreover, according to the ESA, the sail can be increased or reduced to meet the requirements, but it will always be very light and economical compared to the fuel required for a controlled de-orbit maneuver. Although at the end of the mission, or after a failure, the spacecraft may be tipping over, various aerothermodynamic calculations have shown that it is safe to deploy the sail and that the satellite will slowly self-stabilize without the need for an expensive guidance, navigation, and added control system. Another advantage is that the entire subsystem, especially the spark plug membrane, burns out very easily on reentry into the atmosphere, which is important for not falling to Earth.