# Performance Estimate of a Spin-Stabilized Drag Sail for Spacecraft Deorbiting
## 2 Mathematical Model  
Assume that a reference spacecraft in an LEO has reached the end of its operational life. To facilitate the deorbiting phase, a drag sail is deployed at time \(t_{0}\) with the aim of significantly increasing the surface area exposed to atmospheric particle flux. It is useful to introduce a radial-tangential-normal (RTN) reference frame \(\mathcal{F}_{\text{RTN}}\), with unit vectors \(\hat{\mathfrak{z}}_{R},\hat{\mathfrak{z}}_{T},\hat{\mathfrak{z}}_{N}\), centered on the spacecraft's center of mass \(O\). With reference to Figure 1, \(\hat{\mathfrak{z}}_{R}\) is aligned with the Earth-spacecraft direction, \(\hat{\mathfrak{z}}_{N}\) is perpendicular to the plane of the osculating orbit in the direction of the angular momentum vector, and \(\hat{\mathbf{\hat{t}}}_{T}\) lies on the plane of the osculating orbit and completes the right-handed triad. In order to avoid singularities, the system dynamics is conveniently described in terms of modified equinoctial orbital elements (MEOEs) \(\{p,\,f,\,g,\,h,\,k,\,L\}\) of the osculating orbit, defined as [56]  
\[p\triangleq a(1-e^{2})\quad,\quad f\triangleq e\cos(\Omega+ \omega)\quad,\quad g\triangleq e\sin(\Omega+\omega)\quad,\] \[h\triangleq\tan(i/2)\cos\Omega\quad,\quad k\triangleq\tan(i/2 )\sin\Omega\quad,\quad L\triangleq\Omega+\omega+\
u \tag{1}\]  
where \(\{a,\,e,\,i,\,\Omega,\,\omega,\,\
u\}\) are the classical orbital elements of the osculating orbit, with \(a\) the semimajor axis, \(e\) the eccentricity, \(i\) the inclination with respect to the Earth's equatorial plane, \(\Omega\) the right ascension of the ascending node, \(\omega\) the argument of the pericenter, and \(\
u\) the true anomaly. Starting from Equation (1), the classical orbital elements can be expressed as a function of the MEOEs as  
\[a=p/(1-e^{2})\quad,\quad e=\sqrt{f^{2}+g^{2}}\quad,\quad i=2\, \arctan(\sqrt{h^{2}+k^{2}})\quad,\] \[\sin\Omega=k/\sqrt{h^{2}+k^{2}}\quad,\quad\cos\Omega=h/\sqrt{h^{ 2}+k^{2}}\] \[\sin\omega=\frac{gl-fk}{\sqrt{(f^{2}+g^{2})(h^{2}+k^{2})}}\quad, \quad\cos\omega=\frac{fh+gk}{\sqrt{(f^{2}+g^{2})(h^{2}+k^{2})}}\quad,\] \[\
u=L-\Omega-\omega \tag{2}\]  
Using the MEOEs, the dynamics of the spacecraft's center of mass \(O\) in matrix form is  
\[\dot{\mathbf{x}}=\mathbb{A}[\mathbf{a}_{p}]_{\mathcal{S}_{\mathrm{RTN}}}+\mathbf{b} \tag{3}\]  
where \(\mathbf{x}\triangleq[p,\,f,\,g,\,h,\,k,\,L]^{\tau}\) is the state vector, whose initial value \(\mathbf{x}_{0}\triangleq\mathbf{x}(t_{0})\) is known, being obtained from orbital elements at the start of the deorbiting phase, \(\mathbf{a}_{p}\in\mathbb{R}^{3\times 1}\) is the total perturbative (non-Keplerian) acceleration acting on the spacecraft, and superscript \(\tau\) denotes the transpose operator. The matrix \(\mathbb{A}\in\mathbb{R}^{6\times 3}\) in Equation (3) is defined as  
\[\mathbb{A}\triangleq\sqrt{\frac{p}{\mu_{\oplus}}} \begin{bmatrix}0&\frac{2\,p} {1+\cos L+g\sin L}&0\\ \sin L&\frac{(2+f\cos L+g\sin L)\cos L+f}{1+f\cos L+g\sin L}&-\frac{g(h\sin L- k\cos L)}{1+f\cos L+g\sin L}\\ -\cos L&\frac{(2+f\cos L+g\sin L)\sin L+g}{1+f\cos L+g\sin L}&\frac{(h\sin L- k\cos L)}{1+f\cos L+g\sin L}\\ 0&0&\frac{(1+h^{2}+k^{2})\cos L}{2(1+f\cos L+g\sin L)}\\ 0&0&\frac{(1+h^{2}+k^{2})\sin L}{2(1+f\cos L+g\sin L)}\\ 0&0&\frac{h\sin L-k\cos L}{1+f\cos L+g\sin L}\\ \end{bmatrix}  \tag{4}\]  
Figure 1: Sketch of RTN and ECI reference frames. while vector \(\mathbf{b}\in\mathbb{R}^{6\times 1}\) is given by  
\[\mathbf{b}\triangleq \begin{bmatrix}0&0&0&0&0&\sqrt{\mu_{\oplus}\,p}\left(\frac{1+f \cos L+g\sin L}{p}\right)^{2}\end{bmatrix} ^{\intercal} \tag{5}\]  
where \(\mu_{\oplus}\simeq 398,\!600\)\(\,\mathrm{km}^{3}/\mathrm{s}^{2}\) is the Earth's gravitational parameter. A suitable model for the perturbative acceleration vector \(\mathbf{a}_{P}\) is necessary. In this analysis, we include perturbations due to aerodynamic forces, Earth's oblateness, and solar radiation pressure while, since only LEOs are considered, third-body effects and other sources of perturbation are neglected, in agreement with Refs.