# Performance Estimate of a Spin-Stabilized Drag Sail for Spacecraft Deorbiting
## 3 Spinning Drag Sail Dynamics
 \tag{17}\]  
Figure 3: Sketch of the spinning drag sail-based deorbiting. Accordingly, the components of \(\hat{\mathbf{n}}\) in the ECI frame are given by  
\[[\hat{\mathbf{n}}]_{\text{ECI}}\equiv[\hat{\mathbf{n}}_{0}]_{\text{ECI}}=[\hat{\mathbf{o}}_{0 }]_{\text{ECI}}=\mathbb{R}_{0}^{\tau}[\hat{\mathbf{o}}(t_{0})]_{\text{RTN}} \tag{18}\]  
where \(\mathbb{R}_{0}\) is obtained by substituting the MEOEs of the starting orbit into Equation (17), and \([\hat{\mathbf{o}}_{0}]_{\text{RTN}}\triangleq[\mathbf{v}_{0}]_{\text{RTN}}/\|\mathbf{v}_{0}\|\) is calculated from MEOEs by means of Equation (9). Also, the (time-varying) components of \(\hat{\mathbf{n}}\) in the RTN frame are given by  
\[[\hat{\mathbf{n}}]_{\text{RTN}}=\mathbb{R}[\hat{\mathbf{n}}_{0}]_{\text{ECI}}=\mathbb{ R}\,\mathbb{R}_{0}^{\tau}[\hat{\mathbf{o}}_{0}]_{\text{RTN}} \tag{19}\]  
where \(\mathbb{R}\) is obtained at each time instant by substituting the instantaneous values of MEOEs into Equation (17). The aerodynamic effects modeled by Equations (7) and (8) may be conveniently written directly in the RTN frame. To this end, the velocity vector \([\mathbf{v}]_{\text{RTN}}\) is calculated by substituting the instantaneous values of MEOEs into Equation (9), and consequently \([\hat{\mathbf{o}}]_{\text{RTN}}=[\mathbf{v}]_{\text{RTN}}/\|\mathbf{v}\|\) is obtained. Then, the lift unit vector components in \(\mathcal{T}_{\text{RTN}}\) are calculated through Equation (10) as functions of \([\hat{\mathbf{o}}]_{\text{RTN}}\) and \([\hat{\mathbf{n}}]_{\text{RTN}}\), the latter being given by Equation (19). Similarly, the values of drag (\(C_{D}\)) and lift (\(C_{L}\)) coefficients are obtained by Equations (11) and (12). As for the atmospheric density, \(\rho\) could be obtained from MATLAB built-in function atmosnrlmsise00, which is based on the NRLMSISE-00 model [66]. It requires solar flux and geomagnetic data available online (see [[https://celestrak.org/spacedata](https://celestrak.org/spacedata)]([https://celestrak.org/spacedata](https://celestrak.org/spacedata)), retrieved on 20 September 2023) and provides \(\rho\) as a function of the time instant \(t\) and of the spacecraft position vector \(\mathbf{r}\) expressed in the ECI frame, that is,  
\[[\mathbf{r}]_{\text{ECI}}=\mathbb{R}^{\tau}[\mathbf{r}]_{\text{RTN}}=\mathbb{R}^{\tau } \begin{bmatrix}\frac{p}{1+\sqrt{\cos L+g\sin L}}\\ 0\\ 0\end{bmatrix}  \tag{20}\]  
However, since the calculation of \(\rho\) through the NRLMSISE-00 model is computationally demanding, the number of required evaluations of the atmospheric density is reduced by applying an iterative procedure, analogous to what is done in Ref. [59]. More precisely, at a generic time instant \(\widetilde{t}\) at which MEOEs are known, it is possible to calculate the osculating orbit period \(\widetilde{T}=2\,\pi\sqrt{\hat{\mathbf{a}}^{3}/\mu_{\oplus}}\), where the tilde symbol denotes quantities calculated at \(t=\widetilde{t}\). The orbital period \(\widetilde{T}\) is divided into \(N\) equally spaced instants belonging to the set \(\mathcal{S}\) defined as  
\[\mathcal{S}=\left\{\widetilde{t},\,\widetilde{t}+\frac{\widetilde{T}}{N-1}, \,\widetilde{t}+\frac{2\widetilde{T}}{N-1},\ldots,\widetilde{t}+\frac{(N-2) \widetilde{T}}{N-1},\,\widetilde{t}+\widetilde{T}\right\} \tag{21}\]  
and the spacecraft is assumed to move on the osculating (Keplerian) orbit. Its angular position \(L(t)\) at each instant \(t\in\mathcal{S}\) is obtained by solving a [PERSON]'s problem and, knowing \(L\), it is possible to calculate the corresponding state vector \(\mathbf{x}(t)\) and the spacecraft position vector in the ECI frame through Equation (20). The latter, together with the time instants \(t\in\mathcal{S}\), the solar flux and geomagnetic data, are given as inputs to the atmosnrlmsise00 function. The set of atmospheric density values  
\[\mathcal{D}=\left\{\rho(\widetilde{t}),\,\rho(\widetilde{t}+\widetilde{T}/(N -1)),\ldots,\rho(\widetilde{t}+\widetilde{T})\right\} \tag{22}\]  
is therefore obtained.