# Performance Estimate of a Spin-Stabilized Drag Sail for Spacecraft Deorbiting
## 4 Performance Estimate of Spinning Drag Sail
One possible explanation for this counterintuitive result is provided by experimental data obtained from measurements made by LightSail-2, a 3U CubeSat that deorbited from a medium-inclination orbit, according to which the aerodynamic torque can help the spacecraft stabilize in a favorable attitude in the final stages of its decay trajectory [72]. Similar considerations can be extended to the results shown in Table 3, which refer to deorbiting scenarios starting from a solar minimum. In this case, the time savings are generally less substantial than in the solar maximum cases shown in Table 2. However, the decay time reductions achieved with a spinning sail are still relevant, particularly if tighter time limits are to be adopted to mitigate the orbital debris problem in the near future. Again, the only exception is the medium-inclination case, for which no time savings are achieved with a spinning sail. An even more complete picture of the performance of a spinning drag sail is obtained from the graphs showing the temporal changes in altitude for the three different types of drag sails considered. An example of such graphs is shown in Figures 4 and 5, which \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Orbit** & **Sail Configuration** & **Decay Time [Years]** & **Time Saving [\%]** \\ \hline \multirow{3}{*}{dawn-dusk SSO} & three-axis & 0.241 & 66.8 \\  & spinning & 0.477 & 34.4 \\  & uncontrolled & 0.727 & \\ \hline \multirow{3}{*}{noon-midnight SSO} & three-axis & 0.222 & 68.3 \\  & spinning & 0.441 & 37.1 \\  & uncontrolled & 0.702 & \\ \hline \multirow{3}{*}{medium inclination} & three-axis & 0.231 & 67.8 \\  & spinning & 0.777 & \(-8.5\) \\  & uncontrolled & 0.717 & \\ \hline \multirow{3}{*}{equatorial} & three-axis & 0.209 & 73.0 \\  & spinning & 0.407 & 47.5 \\ \cline{1-1}  & uncontrolled & 0.775 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Deorbiting performance for different drag sail configurations and initial orbits under high-solar-activity conditions. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Orbit** & **Sail Configuration** & **Decay Time [Years]** & **Time Saving [\%]** \\ \hline \multirow{3}{*}{dawn-dusk SSO} & three-axis & 2.154 & 42.6 \\  & spinning & 3.103 & 17.2 \\  & uncontrolled & 3.750 & \\ \hline \multirow{3}{*}{noon-midnight SSO} & three-axis & 2.071 & 39.5 \\  & spinning & 2.984 & 12.8 \\  & uncontrolled & 3.424 & \\ \hline \multirow{3}{*}{medium inclination} & three-axis & 2.187 & 36.4 \\  & spinning & 3.434 & 0.1 \\ \cline{1-1}  & uncontrolled & 3.437 & \\ \hline \multirow{3}{*}{equatorial} & three-axis & 2.079 & 38.8 \\ \cline{1-1}  & spinning & 3.093 & 8.9 \\ \cline{1-1}  & uncontrolled & 3.396 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Deorbiting performance for different drag sail configurations and initial orbits under low-solar-activity conditions. illustrate the deorbiting profiles for a dawn-dusk and a noon-midnight SSO, respectively. In both cases, it is evident that the rapidity of altitude variation in the case of uncontrolled drag sail is significantly lower than in the case of spinning sail, especially under conditions of high solar activity. Figure 6 shows the same results as in Figure 5 for a spinning drag sail during the first day of decay, highlighting the short-term oscillations in spacecraft altitude due to the action of perturbative accelerations, which are associated with the spacecraft motion along the osculating orbit. This highly perturbed environment is the reason for the high computational costs of numerical simulations. Figure 4: Decay trajectory from a dawn-dusk SSO for a satellite with parameters given in Table 1, with different drag sail configurations (blue = three-axis stabilized sail; black = spin-stabilized sail; red = uncontrolled sail). Figure 5: Decay trajectory from a noon-midnight SSO for a satellite with parameters given in Table 1, with different drag sail configurations (blue = three-axis stabilized sail; black = spin-stabilized sail; red = uncontrolled sail).