## 4 Results
### Electric Propulsion
This non-linear effect could affect the optimization results toward a higher thrust solution. Technical limits in the total firing time could also have the same consequence. For an electric propulsion system, it is also possible to activate the thruster only for a fraction of the orbit, around the apogee, in order to lower the perigee down to an altitude where the aerodynamic drag prevails and deorbits the spacecraft. This is analogous of the half Hohmann strategy for the chemical propulsion system, providing a propellant mass saving. However, in this case the deorbit is slow, and, for the same thrust, the decay time gets longer. In Figures 23-26, an example is shown of an electric thruster that is activated in an arc of only \(18^{\circ}\) before and after the apogee, so \(36^{\circ}\) in total or 10% of the entire orbit. Figure 23: Propellant mass consumption for an electric propulsion system that is activated only \(+/-18^{\circ}\) around the apogee: (**a**) full picture; (**b**) zoom in. In Figure 23, it is possible to see the propellant consumption with time. In Figure 23b, it is possible to see that the thruster is activated only around 10% of the time. In Figure 23a, it is possible to see that the propellant mass linearly increases with time (albeit at steps) and reaches the value of the half Hohmann transfer strategy. The little discrepancy appears only at low release altitudes as the atmospheric drag adds on the propulsion thrust helping in a further propellant mass saving. In Figure 24 it is possible to see that this partial activation of the electric thruster reduces the energy of the orbit (i.e., its semi-major axis) mainly through a decrease in the perigee altitude. Only when atmospheric drag comes into play does circularization occur. The eccentricity of the orbit increases (remaining very small) with the flight-path angle (Figure 25) until drag-induced circularization occurs. Figure 26 shows two details of the increase in flight-path angle (i.e., eccentricity) forced by the apogee thrusting phases (Figure 26a) and of the final circularization induced by the atmospheric drag (Figure 26b). It is interesting to see how this strategy works for a fixed decay time. In this case, the shortest is the activation time and the highest should be the thrust in order to complete the decay in the same time. Apparently, the thrust should be inversely proportional to the activation time. This is correct as an order of magnitude estimate, but it is necessary to consider that as the activation time is decreased, the energy required to be drawn from the orbit is also cut by almost half. Figure 26: Flight-path angle during orbital decay for an electric propulsion system that is activated only +/ – 18\({}^{\circ}\) around the apogee: (a) initial perigee lowering; (b) drag circularization. Figure 25: Orbital decay for an electric propulsion system that is activated only +/ – 18\({}^{\circ}\) around the apogee: (a) eccentricity; (b) flight-path angle. Consequently, if the thrust is scaled as the inverse of the activation time (Figure 27a), the decay time follows the behavior of Figure 27b. The decay time for short activation times is almost half the one for a continuous operation. As expected, the total thruster actuation time is equal to the total decay time for continuous thrusting and goes to zero as the activation time fraction goes to zero (Figure 28a). This mean that partializing the activation time could help being compliant with thruster firing time limitations. As the eccentricity is very low there is negligible difference between true anomaly, eccentric anomaly and mean anomaly; the same stands for activation time in terms of fraction of orbit or fraction of orbital period (Figure 28b). The mass of the thruster is proportional to the thrust of the system, and so increases as the activation time is reduced (Figure 29a). The propellant mass shifts from the one of a half Hohmann maneuver to the one of a full Hohmann maneuver if the transfer does not reach very low altitudes.