## 2 Deorbiting Technologies
The value of the solar pressure is rather low, around 4.5 \(\mu\)N/m\({}^{2}\). The altitude at which the solar and aerodynamic pressure are equal is around 600 km. At this altitude, the drag sail is already not the best-performing solution, thus the solar sail is not well suited for deorbit, particularly taking into account that the solar vector is not aligned with the opposite of the velocity vector for most of the orbit (unlike aerodynamic drag), thereby dramatically reducing its effectiveness and requiring an active attitude control to properly orient the sail. The classical way to deorbit a satellite is to use a propulsion system. The parameters characterizing the propulsion system are the specific impulse _Isp_, thrust \(T\), total impulse \(I_{tot}\), propellant mass \(m_{p}\). The propellant mass is dependent on the delta-v (\(\Delta v\)) required for deorbiting trough the Tsiolkovsky equation:  
\[\Delta v=Isp\;g_{0}\;\ln\left(m_{i}/m_{f}\right) \tag{3}\]  
\[m_{p}=m_{i}-m_{f} \tag{4}\]  
where the subscripts \(i\) and \(f\) stand for initial and final, respectively. For high-thrust systems, an impulsive maneuver can be considered and a Hohmann transfer from the original orbit to another at lower altitude with corresponding lower lifetime can be performed. \[\Delta v_{H}=(v-v_{a})+(v_{p}-v_{0}) \tag{5}\]  
where \(a\) stands for apogee, \(p\) for perigee, \(H\) for Hohmann and \(v\) is the instantaneous orbital velocity:  
\[v=\sqrt{\mu\left(\frac{2}{r}-\frac{1}{a}\right)} \tag{6}\]  
Which is constant for a circular orbit. \(\mu\) is the Earth gravitational constant, \(r\) the local orbit radius and \(a\) is the semi-major axis. Required delta-v and thrust are generally sufficiently high to force the use of a chemical propulsion system, which are divided between solids, liquids and, less frequently, hybrids. Solid systems have the advantage to be very compact and simple, favoring a dedicated one-shot device [29; 30; 31]. Metallized solid propellants release solid particles, so less energetic particle-free formulations are preferable, particularly for deorbit from high altitudes. Solid rockets have relatively high thrusts, which can be an issue for attitude control; in fact, often the motor requires an active thrust vector control, or the spacecraft needs to be spun during firing. Solid rockets are not suited for multifunctional use. On the contrary, liquids tend to be more complex and bulkier, but can be integrated in order to provide other propulsive functions other than deorbiting (like orbit raising and station keeping). Thrust is relatively low so attitude (and trajectory) control is much easier and precise. Hybrid propulsion has its own peculiarities as described in [32; 33; 34; 35] and has been proposed for active deorbiting of large items [36; 37]. For low-thrust systems, a continuous thrusting phase is necessary. The delta-v is the difference between the original (marked with 0) orbital velocity and the final one. \[\Delta v=v_{0}-v \tag{7}\]  
In LEO, the orbits always have a small altitude compared with the Earth radius and consequently a very low eccentricity. Therefore, the difference between a Hohmann transfer and a continuous transfer becomes negligible. The ratio between the continuous delta-v and the impulsive (Hohmann) one is given by the following equation [38]:  
\[\frac{\Delta v}{\Delta v_{H}}=\left[\sqrt{2\bigg{(}1+\frac{2\sqrt{r/r_{0}}}{r /r_{0}+1}\bigg{)}}-1\right]^{-1} \tag{8}\]  
The ratio is only 0.32% higher than one for a deorbiting from 2000 to 300 km (828.2 m/s vs. 825.6 m/s). It is worth noting that, for altitudes above 1400 km, it is more efficient to move the satellite into a disposal orbit slightly above 2000 km than deorbit it [11]. However, thanks to the exponential behavior of the atmospheric density, instead of lowering all the orbit to a new altitude, it is possible to decrease only the perigee to an altitude slightly below the one necessary for a circular orbit to obtain the same decay time. The first half of a Hohmann transfer is roughly one-half of the total delta-v.