# Performance Estimate of a Spin-Stabilized Drag Sail for Spacecraft Deorbiting
## 4 Performance Estimate of Spinning Drag Sail  
The behavior of a spinning drag sail was simulated in some test scenarios with the aim of estimating the decay time of a spacecraft equipped with such a deorbiting device. For comparison purposes, the same deorbiting scenarios were also simulated for a three-axis stabilized drag sail [68] and for an uncontrolled drag sail. In all simulations, a total spacecraft mass \(m=15\) kg and a drag sail area of \(5\,\text{m}^{2}\) were assumed. The chosen total mass is compatible with a 12U CubeSat, representative of small satellites whose numbers are increasing and which pose a greater deorbiting challenge than other common (and smaller) configurations, such as the 1U or 3U CubeSats. The value of the sail area is in accordance with drag sails currently being developed for small and medium-sized spacecraft [64]. The characteristics of the satellite used in the test case scenarios are summarized in Table 1.  
The dynamics of a three-axis stabilized drag sail is obtained by integrating Equation (3) and evaluating the perturbative accelerations through Equations (7), (8), (13) and (14). However, in this case, it is assumed that the drag sail constantly exposes its entire surface to the flow of atmospheric particles by means of a perfectly functioning three-axis stabilization system or a suitable passively stabilizing drag sail design [44; 46; 47; 48]. This amounts to assuming that \(\mathbf{\hat{n}}\equiv\widetilde{\sigma}\) at each time \(t\geq t_{0}\) during the deorbiting trajectory. Such a drag sail configuration provides the shortest decay times but requires a fully functional attitude control system. In the uncontrolled case, instead, the drag sail is again deployed by the spacecraft at \(t_{0}\), but it is assumed that the spacecraft does not have a functioning attitude control system, so it is unable to control its attitude. Therefore, it is subject to environmental torques and undergoes a tumbling motion during orbital decay. In this case, the dynamics of the spacecraft's center of mass is still obtained by integrating Equation (3), but the computation of the components of \(\mathbf{\hat{n}}\) requires some further clarification. In fact, \(\mathbf{\hat{n}}\) is first \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Total spacecraft mass \(m\) & 15 & kg \\ \hline Spacecraft body mass & 14.5 & kg \\ \hline Drag sail mass & 0.5 & kg \\ \hline Spacecraft body dimensions & 0.4 \(\times\) 0.3 \(\times\) 0.1 & m\({}^{3}\) \\ \hline Drag sail area \(A\) & 5 & m\({}^{2}\) \\ \hline Moments of inertia & \{0.5375, 0.5298, 0.4331\} & kg m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Spacecraft design parameters used in the simulations. Drag sail characteristics are taken from the ADEO-N model, see Ref. [64]. given in a body-fixed reference frame. Then, it is transformed into \(\mathcal{F}_{\text{RTN}}\) by applying a rotation matrix whose components depend on the Euler angles between the body-fixed frame and the RTN frame. These angles are obtained by modeling the spacecraft attitude by means of an attitude quaternion [69, 70, 71] and calculating the angular velocity of the spacecraft with respect to \(\mathcal{F}_{\text{RTN}}\) by means of the Euler equations. Since these equations need the spacecraft's moments of inertia, the latter were calculated by accounting for the contributions of the spacecraft body, modeled as a uniform parallelepiped of 14.5 kg with dimensions 0.4 m \(\times\) 0.3 m \(\times\) 0.1 m, and a square, flat, uniform sail membrane with a mass of 0.5 kg. The corresponding principal moments of inertia are given in Table 1. Finally, the external torques acting on the spacecraft were calculated in analogy with Refs. [46, 47] by taking into account three contributions, that is, the torque generated by aerodynamic effects \(\mathbf{a}_{P_{D}}+\mathbf{a}_{P_{L}}\), the torque due to solar radiation pressure, and the gravity gradient torque.