## 4 Results
### Natural Decay  
First of all, as a reference, it is important to determine the natural decay of a typical satellite into LEO without the use of a specific device.  
The decay time mainly follows the (inverse) behavior of density with altitude, so it is exponentially longer as the altitude increases (Figure 9). Moreover, the natural decay accelerates as the altitudes decreases, so the majority of the time is spent near the original altitude and the majority of the fall happens near the end of the decay time (Figure 10). This means also that in the case of deorbiting with a special device like a chemical thruster, the majority of the reduction in decay time is obtained in the initial part of the altitude decrease.  
Figure 8: Natural orbital decay from an initial elliptical orbit (275 \(\times\) 700 km) calculated through the numerical integration of the equations of motion: (**a**) flight-path angle; (**b**) orbits’ shape.  
Figure 9: Natural decay time as a function of altitude, and parametric with spacecraft mass-to-area ratio: (**a**) lower altitudes; (**b**) higher altitudes, black line is the 25 years limit.  
The decay time is also linearly dependent on the ballistic coefficient, which is mainly determined by the ratio between the mass and the frontal area of the satellite. This ratio depends on the scale and the design of the system. For the same, identical satellite (shape and density), the scale should change the ballistic coefficient as the volume to ratio, i.e., a power of 3/2 times the linear size of the system. However, small satellites have typically higher density (around 1000-2000 kg/m\({}^{3}\) for a cubesat) than the larger ones (even below 100 kg/m\({}^{3}\), like the Hubble Space telescope), which mitigates the scale effect. Thus, the majority of satellites have a mass-to-area ratio between 100 and 500 kg/m\({}^{2}\) (0.01 to 0.002 m\({}^{2}\)/kg). Anyway, specific designs with large, deployable surfaces can have very low ballistic coefficients, while in the opposite circumstances other designs elongated in the direction of the flow aimed at low-altitude flying (like GOCE [48]) can have a particularly high ballistic coefficient.  
From Figure 9, it is possible to see that the 25-year limit is achieved around 500-650 km, while a re-entry in less than five years necessitates an altitude below 300-400 km. Figures 9 and 10 consider circular orbits and the natural decay corresponds to a spiraling down of the satellite trajectory where the eccentricity remains negligible.  
The almost exponential trend of the atmospheric density with altitude has a dramatic influence on deorbiting behavior. In the case of an elliptical orbit, the peak of drag at the perigee induces a strong reduction of the apogee altitude in a process of circularization. When the eccentricity becomes very small, the perigee starts to fall as the apogee and the orbit begin spiraling down.  
The deorbiting time depends on the orbital energy and the energy dissipation. The first parameter is related to the orbit semi-major axis, while the second is strongly related with the minimum (i.e., perigee) altitude.  
In Figure 11 it is possible to see that the same spacecraft has nearly the same decay time (around 2 months) for a more energetic orbit with a lower perigee (\(a=R_{t}+450\) km, \(h_{p}=200\) km, \(h_{a}=700\) km) and a less energetic circular orbit (\(h=316\) km). This is due to the fact that the first, more energetic orbit experiences a larger energy dissipation driven by the higher drag at the (lower) perigee.  
Figure 10: Natural decay of a satellite with time, and parametric with spacecraft mass-to-area ratio: (**a**) 300 km initial altitude; (**b**) 450 km initial altitude.