# Performance Estimate of a Spin-Stabilized Drag Sail for Spacecraft Deorbiting
## 4 Performance Estimate of Spinning Drag Sail
For simulation purposes, the center of pressure of the aerodynamic torques was assumed to coincide with that of the solar radiation pressure torques, which are located on the sail membrane and are displaced by 0.1 m relative to the center of the sail in each of the two spatial coordinates. To make a comparison between different drag sail configurations, several initial orbits were considered, chosen from the most populated orbital ranges, see Ref. [10]. All of the selected orbits are circular, with altitude \(h_{0}=600\) km, that is, the initial semimajor axis is \(a_{0}=6978\) km. This choice is motivated by a trade-off between the importance of an effective deorbiting strategy (which increases with altitude due to the reduced atmospheric density) and computational effort, which is reduced for shorter flight times. The four selected orbits are (i) a dawn-dusk Sun-synchronous orbit (SSO), (ii) a noon-midnight SSO, (iii) a medium-inclination orbit with \(i_{0}=53\) deg and \(\Omega_{0}=30\) deg, and (iv) an equatorial orbit. These choices are based on an ESA report [10]. In particular, Sun-synchronous orbits (i) and (ii) are highly populated by spacecraft performing remote sensing observation, whereas the 53 deg-inclination case (iii) has been chosen because of the large density of satellite constellations [9], thus increasing the collision probability for nearby spacecraft. The equatorial orbit (iv) has been included to account for a low-inclination scenario. For each selected orbit, the deployment is assumed to take place when \(L_{0}=\Omega_{0}\), without loss of generality. For each initial orbit, two simulations were performed, selecting \(t_{0}\) either on 1 January 2014 or on 1 January 2019. The first case is near a solar maximum (reported in the last months of 2014) and is representative of the most favorable situation to achieve a fast decay. Starting in January 2019, on the other hand, is equivalent to performing much of the deorbiting within a solar activity minimum, thus achieving slow orbital decay. Numerical simulations were performed with commercial MATLAB software (version R2023a) on a desktop computer with an i7-11700F-2.50 GHz processor. Considering the three-axis stabilized and spinning sail configurations, typical simulation times are in the order of minutes when starting at a solar maximum, and in the order of tens of minutes if \(t_{0}\) is at a solar minimum. The uncontrolled satellite case, on the other hand, involves a significant computational cost, such that each numerical simulation requires a computational time in the order of several hours starting from a solar maximum and as much as 15-20 h starting from a solar minimum. Such a significant difference in computational cost for the uncontrolled scenario is mainly due to the need to simulate the dynamics of spacecraft attitude, which has much faster dynamics than the evolution of MEOEs and thus requires short time steps to be properly evaluated. The results of the numerical simulations are given in Tables 2 and 3 for a deorbiting starting at a solar maximum and at a solar minimum, respectively. The two tables show the decay times from the initial altitude of 600 km to a final altitude set equal to 200 km, a value chosen assuming that reentry ends when the spacecraft reaches the dense layers of the atmosphere. In addition, the last column of the tables shows the time savings obtained with a three-axis stabilized sail and a spinning sail compared to the uncontrolled case. Table 2 shows that for deorbiting from a high solar activity, the time savings obtained with a spinning sail are significant, although lower than those given by a three-axis stabilizedsail, and are particularly remarkable for Sun-synchronous orbits, which are very common for LEO satellites. The only notable exception is the case of medium-inclination orbit, for which no time savings are achieved with a spinning sail.