# Solar Sail Transfer Trajectory Design for Comet 29P/Schwassmann-Wachmann 1 Rendezvous
## 2 Mathematical Preliminaries and Models
### Trajectory Optimization  
For an assigned value of the characteristic acceleration \(a_{c}\), the optimal orbit-to-orbit transfer has been studied using an indirect approach [38] in which the Hamiltonian function \(\mathcal{H}\) is obtained from Equation (10) as [25]  
\[\mathcal{H}\triangleq\dot{\mathbf{x}}\cdot\mathbf{\lambda}=\left(\mathbb{A}\,\mathbf{a} \right)\cdot\mathbf{\lambda}+\mathbf{d}\cdot\mathbf{\lambda} \tag{15}\]  
where \(\mathbf{\lambda}\triangleq\left[\lambda_{p},\,\lambda_{f},\,\lambda_{g},\, \lambda_{h},\,\lambda_{k},\,\lambda_{L}\right]^{\intercal}\) is the adjoint vector [39]. The time derivative of \(\lambda\) is given by the classical Euler-Lagrange equations  
\[\dot{\mathbf{\lambda}}=-\frac{\partial\mathcal{H}}{\partial\mathbf{x}}=-\frac{ \partial(\mathbb{A}\,\mathbf{a}\cdot\mathbf{\lambda})}{\partial\mathbf{x}}-\frac{ \partial(\mathbf{d}\cdot\mathbf{\lambda})}{\partial\mathbf{x}} \tag{16}\]  
which gives six first-order, non-linear, differential equation in the variables \(\{\lambda_{p},\,\,\lambda_{f},\,\,\lambda_{g},\,\,\lambda_{h},\,\,\lambda_{k},\,\,\lambda_{L}\}\), whose explicit expressions are omitted here for the sake of brevity.  
During the transfer, the optimal control vector, that is, the optimal sail attitude in terms of time variation of \(\hat{\mathbf{n}}\), is obtained by using [PERSON]'s maximum principle, that is, by finding the value of the normal unit vector that maximizes (at any time) the Hamiltonian function given by Equation (15). In this respect, a numerical approach is used to calculate the two (independent) scalar variables that maximize the value of \(\mathcal{H}\) and the procedure is illustrated in the recent authors' work [25].  
The equations of motion (10) and the Euler-Lagrange Equation (16) form the two-point boundary problem (TPBVP) which is completed by 13 boundary conditions, because the (optimal) flight time \(t_{f}\) is an output of the optimization process. In particular, five (initial) conditions are given by Equation (13), while five (final) conditions are obtained by enforcing the rendezvous with comet SW1 orbit. In this respect, using the data from Table 1 and Equation (2), one obtains  
\[p(t_{f})=p_{f}\triangleq 6.0379\,\mathrm{au}\,\,,\,f(t_{f})=f_{ f}\triangleq 4.4425\times 10^{-2}\,\,,\,g(t_{f})=g_{f}\triangleq 2.996\times 10^{-3}\,\,,\] \[h(t_{f})=h_{f}\triangleq 5.5214\times 10^{-2}\,\,,\,k(t_{f})=k_{ f}\triangleq-6.0465\times 10^{-2} \tag{17}\]  
Finally, the last three scalar constraints required to complete the TPBVP are obtained from the transversality condition [39] as  
\[\lambda_{L}(t_{0})=\lambda_{L_{0}}\triangleq 0\,\,,\,\lambda_{L}(t_{f})= \lambda_{L_{f}}\triangleq 0\,\,,\,\,\mathcal{H}(t_{f})=\mathcal{H}_{f}\triangleq 1 \tag{18}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Force Model** & \(\mathbf{b_{1}}\) & \(\mathbf{b_{2}}\) & \(\mathbf{b_{3}}\) \\ \hline Ideal & 0 & 1 & 0 \\ Optical & 0.0951 & 0.9109 & \(-0.006\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Normalized force coefficients for an ideal and an optical force model, assuming a (rigid and flat) sail membrane with a highly reflective aluminum-coated front side and a highly emissive chromium-coated back side.  
The TPBVP has been solved using an in-house procedure based on the multiple shooting method and the use of genetic algorithms. In particular, the differential equations have been integrated in double precision using a variable order Adams-Bashforth-Moulton solver [40] with absolute and relative errors of \(10^{-10}\). The results of this numerical optimization process are illustrated in the next section, which contains both a parametric analysis of the orbit-to-orbit transfer as a function of the value of the characteristic acceleration and a detailed case study for a specific value of \(a_{c}\).