## 3 Materials and Methods  
In order to compare the different technologies, it is necessary to calculate the de-orbiting behavior. For this a simple Hohmann transfer Equation (5) is used. For a half Hohmann transfer only the first term is necessary. Propellant mass is calculated with Equations (3) and (4). Time for the maneuver can be calculated by simple orbital mechanics [45; 46; 47] as half the orbital period \(T\):  
\[T=2\pi\sqrt{a^{3}/\mu} \tag{11}\]  
but is definitely negligible (around 90 min per each full orbit, the number of orbits depending on the number of impulses that composes the total impulse required) with respect to the typical required deorbiting time. For the calculation of deorbiting time with drag and small forces, the equation of motion is integrated with time:  
\[\ddot{\mathbf{r}}+\frac{\mu}{r^{3}}\mathbf{r}=\mathbf{a}_{F} \tag{12}\]  
where \(\mathbf{a}_{F}\) is the force-induced acceleration (\(F/m\)). This is the so-called Cowell approach. It requires a small timestep in order to avoid a large error buildup. The vectorial equation is integrated with an adaptive step 4 th-5 th-order Runge-Kutta scheme. The error tolerance has been set by repeating the simulation until the relative error between the last two was well below 1%. For the sake of simplicity, the rotation of the atmosphere has been neglected in this work so that the relative velocity is equal to the spacecraft absolute velocity. Two semi-analytical techniques have been also used to drastically reduce the computational time. In case of an initial circular orbit and a slow spiraling down the following equation has been used:  
\[r\approx\frac{r_{0}}{\left(1-\frac{F}{m}\frac{t}{t_{0}}\right)^{2}} \tag{13}\]  
where \(t\leq T\) is the time considered for integration. For elliptical orbits the following approach has been used instead. We assume that \(a\) is almost constant during one orbit:  
\[a=\frac{2a^{2}}{\mu}dE\rightarrow\Delta a_{rev}=\frac{2a^{2}}{\mu}\Delta E_{rev} \tag{14}\]  
The energy decay during one orbit is equal to the work of the forces applied to the spacecraft:  
\[\Delta E_{rev}=\int_{0}^{T}F/mvdt \tag{15}\]  
Figure 2: Variation of the tether force with orbital inclination (10 kg mass device). Which in case of drag:  
\[F/m=-D/m=-0.5\rho v^{2}/B_{c} \tag{16}\]  
where \(B_{c}\) is the ballistic coefficient:  
\[B_{c}=m/(c_{d}A) \tag{17}\]  
Substituting in Equations (14) and (15):  
\[\Delta a_{rev}=-\frac{a^{2}}{B_{c}\mu}\int_{0}^{T}\rho v^{2}vdt \tag{18}\]  
The instantaneous velocity is:  
\[v^{2}=\sqrt{v_{r}^{2}+v_{l}^{2}}=\frac{n^{2}a^{2}}{1-e^{2}}\Big{(}1+e^{2}+2 ecos \theta\Big{)}\approx n^{2}a^{2}(1+2 ecos\theta) \tag{19}\]  
where \(r\) means radial, \(t\) tangential, \(e\) is the eccentricity and \(\theta\) the true anomaly and:  
\[\ddot{\mathbf{r}}+\frac{\mu}{r^{3}}\mathbf{r}=\mathbf{a}_{F} \tag{12}\]0  
As the eccentricity is small the higher terms on \(e\) have been neglected, considering:  
\[\ddot{\mathbf{r}}+\frac{\mu}{r^{3}}\mathbf{r}=\mathbf{a}_{F} \tag{12}\]1  
where \(E\) is the eccentric anomaly. Substituting in Equation (18):  
\[\ddot{\mathbf{r}}+\frac{\mu}{r^{3}}\mathbf{r}=\mathbf{a}_{F} \tag{12}\]2  
Also considering Equation (20):  
\[\ddot{\mathbf{r}}+\frac{\mu}{r^{3}}\mathbf{r}=\mathbf{a}_{F} \tag{12}\]3  
Remembering that:  
\[\ddot{\mathbf{r}}+\frac{\mu}{r^{3}}\mathbf{r}=\mathbf{a}_{F} \tag{12}\]4  
We obtain:  
\[\ddot{\mathbf{r}}+\frac{\mu}{r^{3}}\mathbf{r}=\mathbf{a}_{F} \tag{12}\]5  
Using a simple exponential atmospheric model:  
\[\ddot{\mathbf{r}}+\frac{\mu}{r^{3}}\mathbf{r}=\mathbf{a}_{F} \tag{12}\]6  
where \(h\) is the altitude and \(H\) the scale height.