# Solar Sail Transfer Trajectory Design for Comet 29P/Schwassmann-Wachmann 1 Rendezvous
## 2 Mathematical Preliminaries and Models
### Spacecraft Dynamics  
Paralleling the approach employed by the authors in the recent literature [25], the motion of the solar sail-based spacecraft during the heliocentric transfer is described using the modified equinoctial elements (MEOEs) introduced by [PERSON] in 1985 [28; 29]. In particular, MEOEs \(\{p,\ f,\ g,\ h,\ k,\ L\}\) are elements valid for a generic osculating orbit, which are related to the classical orbital elements \(\{a,\ e,\ i,\ \omega,\ \Omega,\ \
u\}\) through the simple relationships [28; 29]  
\[p=a\Big{(}1-e^{2}\Big{)}\,\ f=e\,\cos(\omega+\Omega)\,\ g=e\,\sin(\omega+ \Omega)\,\ h=\tan(i/2)\,\cos\Omega\,\\ k=\tan(i/2)\,\sin\Omega\,\ L=\
u+\Omega+\omega \tag{2}\]  
while the inverse passage is obtained by using the equations  
\[a=\frac{p}{1-f^{2}-g^{2}} \tag{3}\]  
\[e=\sqrt{f^{2}+g^{2}} \tag{4}\]  
\[i=2\,\arctan\sqrt{h^{2}+k^{2}} \tag{5}\]  
\[\sin\omega=g\,h-f\,k\quad,\quad\cos\omega=f\,h+g\,k \tag{6}\] \begin{table}
\begin{tabular}{c c c c c c} \hline
**Body** & \(a\) **[au]** & \(e\) & \(i\) **[deg]** & _\(\omega\) **[deg]** & _\(\Omega\) **[deg]** \\ \hline Earth & 1.001 & \(1.594\times 10^{-2}\) & \(3.022\times 10^{-3}\) & \(302.978\) & \(159.864\) \\ Comet SW1 & 6.0499 & \(4.4526\times 10^{-2}\) & \(9.3619\) & \(51.4572\) & \(312.4009\) \\ \hline \end{tabular}
\end{table} 
Table 1: Classical orbital elements of the parking (Earth) and target (comet SW1) orbit used in the numerical simulations. \[\sin\Omega =k\quad,\quad\cos\Omega=h \tag{7}\] \[\
u =L-\Omega-\omega. \tag{8}\]  
Note that \(p\) is the semilatus rectum of the spacecraft osculating orbit, see the first of Equation (2), while the true longitude \(L\) is the fast varying variable which indicates the spacecraft angular position along the transfer orbit; see the last of Equation (2). According to [PERSON] [30], if one defines the spacecraft state vector \(\boldsymbol{x}\) as  
\[\boldsymbol{x}\triangleq[p,\;f,\;g,\;h,\;k,\;L]^{\top} \tag{9}\]  
the spacecraft dynamics can be described by the following (compact) differential vectorial equation  
\[\dot{\boldsymbol{x}}=\mathbb{A}\;\boldsymbol{a}+\boldsymbol{d} \tag{10}\]  
where \(\boldsymbol{a}\in\mathbb{R}^{3\times 1}\) is the solar sail-induced propulsive acceleration vector, while \(\mathbb{A}\in\mathbb{R}^{6\times 3}\) and \(\boldsymbol{d}\in\mathbb{R}^{6\times 1}\) are defined as  
\[\mathbb{A}\triangleq \begin{bmatrix}0&\frac{2\;p}{1+f\;\cos L+g\;\sin L}\; \sqrt{\frac{p}{\mu_{\odot}}}&0\\ \sin L\;\sqrt{\frac{p}{\mu_{\odot}}}&\frac{(2+f\;\cos L+g\;\sin L)\;\cos L+f}{1 +f\;\cos L+g\;\sin L}\;\sqrt{\frac{p}{\mu_{\odot}}}&-\frac{g\left(h\;\sin L- k\;\cos L\right)}{1+f\;\cos L+g\;\sin L}\;\sqrt{\frac{p}{\mu_{\odot}}}\\ -\cos L\;\sqrt{\frac{p}{\mu_{\odot}}}&\frac{(2+f\;\cos L+g\;\sin L)\;\sin L+g}{ 1+f\;\cos L+g\;\sin L}\;\sqrt{\frac{p}{\mu_{\odot}}}&\frac{f\left(h\;\sin L- k\;\cos L\right)}{1+f\;\cos L+g\;\sin L}\;\sqrt{\frac{p}{\mu_{\odot}}}\\ 0&0&\frac{\left(1+h^{2}+k^{2}\right)\;\cos L}{2\left(1+f\;\cos L+g\;\sin L \right)}\;\sqrt{\frac{p}{\mu_{\odot}}}\\ 0&0&\frac{\left(1+h^{2}+k^{2}\right)\;\sin L}{2\left(1+f\;\cos L+g\;\sin L \right)}\;\sqrt{\frac{p}{\mu_{\odot}}}\\ 0&0&\frac{h\;\sin L-k\;\cos L}{1+f\;\cos L+g\;\sin L}\;\sqrt{\frac{p}{\mu_{ \odot}}}\end{bmatrix}  \tag{11}\]  
\[\boldsymbol{d}\triangleq \begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0\\ 0\end{bmatrix}  \tag{12}\]  
The six scalar initial conditions associated with the vectorial differential Equation (10) model the spacecraft motion along the Earth's heliocentric orbit at time \(t_{0}\).