## 2 Deorbiting Technologies  
The simplest way to deorbit a satellite in LEO is using the natural aerodynamic decay. This is a passive and inexpensive solution. However, the drag \(D\) produced is linearly dependent on the atmospheric density \(\rho\):  
\[D=0.5\rho v^{2}c_{d}A \tag{1}\]  
The drag coefficient \(c_{d}\) is considered near 2 for a typical satellite (2.2 used in this paper), where \(A\) is the frontal area respect to the incoming flow (the opposite of the velocity vector). Here the density has been calculated with the US Standard Atmosphere 1976 model, _USSA1976_[22]. This model does not take into account temporal and horizontal spatial variations, but only the average behavior with altitude. More complete models are available [23], and it is important to remember that the actual density at a certain altitude can change significantly due to variations in Sun activity. However, once this caveat is known, the current model anyway is considered sufficient to be used for the relative comparison of the different deorbiting technologies. The atmospheric density has roughly an exponential behavior with altitude. Therefore, the lifetime of a dead satellite is also an exponential function of the altitude. Current legislations foresee a decay time below 25 years for non-operational satellites. It is possible to see that the corresponding limit altitude is around 600 km. It is worth remembering that the peak of debris and the current favorite altitude is slightly above this value. However, from the current analyses and trends it is possible to predict (and also to encourage) that sooner or later the legislation should become stricter, reducing the time allowed to complete deorbit. In this case the limit altitude will decrease significantly. The author proposes the possibility to put a mandatory fee on deorbit, with the fee proportional to the deorbit time, or time spent in orbit after end of life in case of failed deorbit (forcing an active debris removal in the latter). In order to improve the decay time, it is possible to add a drag sail [24; 25; 26; 27; 28]. This device is deployed at the end of life to drastically increase the frontal area and consequently the drag, therefore reducing the deorbit time. The device needs an actuation system, i.e., a mechanism to deploy the sail correctly and a communication link or an internal computer that starts the procedure when needed. Consequently, at least two failure points are introduced. The mass of the drag sail is approximated as a linear function of its area. Typical values of area density are around 75 g/m\({}^{2}\) with up to dozens of m\({}^{2}\) of area [17]. After deployment,the satellite will theoretically passively deorbit. However, in the general case the attitude of the satellite could change with time and the drag sail could not present its full area against the flow, reducing its efficacy. Three possibilities are then possible: to accept a performance degradation, but this choice is not preferable as it comes with a significant uncertainty; to add an attitude control during re-entry, perhaps using one already present in the satellite (which, however should cope with different forces compared to its original design requirements); or, finally, find a suitable sail shape design that would preserve its positional stability with respect to the incoming flow. The last possibility seems the most promising, even if not fully developed. However, the drag sail linearly decreases the decay time as the inverse of its frontal area. Therefore, it performs very well at low altitudes, but rapidly loses effectiveness at higher altitudes. A similar solution is the solar sail. The solar sail is a proposed solution to impart a delta-v on a spacecraft without consuming propellant, but rather using the force from solar radiation. Very interesting for certain missions, it can be potentially used also for deorbiting a spacecraft. The solar pressure is:  
\[p_{rad}=I/c \tag{2}\]  
where \(I\) is the solar irradiance and \(c\) the speed of light [8].