# Performance Estimate of a Spin-Stabilized Drag Sail for Spacecraft Deorbiting
## 3 Spinning Drag Sail Dynamics  
The decay trajectory of a drag sail spacecraft can be obtained by integrating Equation (3) with a numerical propagator. As discussed above, the perturbative acceleration of Equation (6) can be calculated as a function of the unit normal vector \(\hat{\mathbf{n}}\) affecting aerodynamic perturbations, see Equations (10)-(12), and the acceleration due to solar radiation pressure, see Equations (14) and (15). In this analysis, the drag sail is assumed to be spinning. The deployment occurs at \(t_{0}\), when the deorbiting phase begins and the spacecraft's angular velocity is perpendicular to the sail surface. The spacecraft rotation generates a centrifugal force that helps to properly deploy the sail, while its residual rotational velocity keeps the drag sail stretched without requiring large structural elements and provides gyroscopic stiffness. As a result, the spinning sail is assumed to keep its angular velocity vector normal to the sail surface (and so the unit vector \(\hat{\mathbf{n}}\)) inertially fixed. A conceptual sketch of the deorbiting scenario is shown in Figure 3.  
Figure 2: Conceptual sketch of cylindrical-shadow eclipse model (not to scale). The procedure for calculating the components of \(\hat{\mathbf{n}}\) is as follows. The angular velocity of the spacecraft is aligned with the orbital velocity at \(t_{0}\) and the deployment occurs when the flight path angle is zero. Without loss of generality, this amounts to assuming that \(\hat{\mathbf{n}}(t_{0})\equiv\hat{\mathbf{v}}(t_{0})\), since \(\hat{\mathbf{n}}\) can be selected in both directions perpendicular to the drag sail surface, see Section 2.1. Since \(\hat{\mathbf{n}}\) is fixed inertially, it is convenient to write the components of \(\hat{\mathbf{n}}(t_{0})\) in an inertial reference frame, as they will remain constant during the deorbiting phase, assuming that the sail spin rate provides a large gyroscopic stiffness. For this purpose, introduce an Earth-centered inertial (ECI) reference system \(\mathcal{T}_{\text{ECI}}\), with origin \(G\) in the Earth's center of mass, and unit vectors \(\hat{\mathbf{i}}\), \(\hat{\mathbf{j}}\), and \(\hat{\mathbf{k}}\). By definition, \(\hat{\mathbf{i}}\) lies on the Earth's equatorial plane and points towards the Aries point \(\hat{\mathbf{\gamma}}\), \(\hat{\mathbf{k}}\) is aligned with the Earth's rotation axis in the direction of the North Pole, and \(\hat{\mathbf{j}}\) belongs to the Equatorial plane and completes the right-handed triad; see Figure 1.  
Since the perturbative acceleration is most simply written in the RTN frame, it is necessary to define the rotation matrix \(\mathbb{R}\) between the ECI frame and the RTN frame, that is,  
\[\mathbb{R} = \begin{bmatrix}\cos u&-\sin u&0\\ \sin u&\cos u&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}1&0&0\\ 0&\cos i&-\sin i\\ 0&\sin i&\cos i\end{bmatrix}\begin{bmatrix}\cos\Omega&-\sin\Omega&0\\ \sin\Omega&\cos\Omega&0\\ 0&0&1\end{bmatrix} =\] \[= \begin{bmatrix}\cos\Omega\cos u-\sin\Omega\cos i\sin u&\sin \Omega\cos u+\cos\Omega\cos i\sin u&\sin i\sin u\\ -\cos\Omega\sin u-\sin\Omega\cos i\cos u&\cos\Omega\cos i\cos u-\sin\Omega \sin u&\sin i\cos u\\ \sin\Omega\sin i&-\cos\Omega\sin i&\cos i\end{bmatrix}  \tag{16}\]  
where \(u\triangleq\omega+\
u\). The same rotation matrix can be expressed in terms of MEOEs by substituting Equation (2) into Equation (16) and performing some algebraic manipulation. The result is  
\[\mathbb{R}= \begin{bmatrix}\frac{\hslash^{2}\cos L+2\hslash k\sin L-k^{2}\cos L +\cos L}{1+k^{2}+k^{2}}&\frac{-\hslash^{2}\sin L+2\hslash k\cos L+\hslash k^ {2}\sin L+\sin L}{1+k^{2}+k^{2}}&\frac{2\hslash\sin L-2\slash k\cos L}{1+k^ {2}+k^{2}}\\ \frac{-\hslash^{2}\sin L+2\hslash k\cos L+k^{2}\sin L-\sin L}{1+k^{2}+k^{2} }&\frac{-\hslash^{2}\cos L-2\hslash k\sin L+\hslash^{2}\cos L+\cos L}{1+k^ {2}+k^{2}}&\frac{2\hslash\cos L+2\slash k\sin L}{1+k^{2}+k^{2}\hslash^{2}} \\ \frac{2\hslash}{1+k^{2}+k^{2}}&-\frac{2\hslash}{1+k^{2}+k^{2}}&\frac{1-\hslash ^{2}-k^{2}}{1+k^{2}+k^{2}}\end{bmatrix}