## 4 Results
### Chemical Propulsion  
With a chemical propulsion system, it is possible to perform a Hohmann transfer from the original orbit to a parking orbit with a defined natural decay time. The mass of the propulsion systems mainly depends on the propellant mass that in turn follows the delta-v between the two orbits. Chemical propulsion has the advantage of being a very fast disposal option, with the Hohmann transfer requiring half an orbit (around 45 min in LEO) for a single impulsive maneuver, or several hours in case the perigee lowering and the final orbit circularization are divided in several burns due to limitation in thrust (and consequently maximum time spent at the apsides). Figures 13-18 have been computed with \(k\) = 0.12 and \(\mathit{Isp}\) = 300 s. For delta-v up to 500 m/s and \(\mathit{Isp}\) = 300 s, propellant mass is almost linear with delta-v (10% error). Chemical propulsion allows satellites to deorbit from any altitude; however, the required mass can be significant, and easily represent more than 10% of the total satellite mass. Figure 13: Deorbiting with chemical propulsion with a full Hohmann transfer, parametric with initial altitude (400–1200 km). Propulsion system mass fraction vs. decay time. Figure 16: Deorbiting with chemical propulsion with a half Hohmann transfer, parametric with decay time: (**a**) delta-\(v\); (**b**) propulsion system mass fraction. Figure 14: Deorbiting with chemical propulsion with a full Hohmann transfer, parametric with decay time: (**a**) delta-\(v\); (**b**) propulsion system mass fraction. Figure 15: Deorbiting with chemical propulsion with a half Hohmann transfer, parametric with initial altitude (400–1200 km): (**a**) decay time vs. perigee altitude; (**b**) propulsion system mass fraction vs. decay time. In order to reduce this mass, it is possible to increase the altitude of the parking orbit, with the consequence of a longer total deorbiting time. However, as it can be seen in Figures 13 and 14, the system mass has a low sensitivity to the total decay time, as the latter changes tremendously for a rather small variation in the final release altitude, thus not affecting much the propulsion system mass (the cases where the final and initial orbits are near, and where the propulsion system mass is small, are excluded). As already pointed out earlier, a more efficient possibility is to simply decrease the perigee of the original orbit with a half Hohmann transfer up to a point where the decay time corresponds to the target one (Figures 15 and 16). In this case, contrary to the circular orbit disposal, the decay time is not only dependent on the final (perigee) release altitude but also on the initial one (which remains the apogee altitude), as can be seen in Figure 15a. Thanks to the exponential behavior of atmospheric density, the perigee of the new orbit is not much lower than the corresponding circular orbit (Figure 17). Consequently, the total delta-v and mass are only slightly higher than half the Hohmann transfer to the circular orbit (around 60%), guaranteeing a significant mass saving. The total mass is thus almost always below 10% of the satellite mass. The qualitative behavior remains the same as before, as it can be seen in Figure 18. The only advantage of the circular orbit disposal is that the satellite is immediately placed far outside the original orbit, while with the more efficient elliptic disposal, the apogee slowly descends from the initial altitude. The mass model considered for this paper is very simplified. In reality the chemical propulsion system mass is a sublinear function of the propellant mass, particularly for  
Figure 17: Deorbiting with chemical propulsion, full Hohmann vs. half Hohmann, decay time vs. release altitude; 800 km initial altitude. Figure 18: Deorbiting with chemical propulsion, full Hohmann vs. half Hohmann, 800 km initial altitude: (**a**) delta-v; (**b**) propulsion system mass fraction. small sizes and lower propellant masses.