# Solar Sail Transfer Trajectory Design for Comet 29P/Schwassmann-Wachmann 1 Rendezvous
## 4 Two-Dimensional Simplified Approach  
The (very) small value of the eccentricity of both the Earth and comet SW1 heliocentric orbit, together with the small relative orbital inclination (slightly above 9 deg, see data in Table 1), suggest the use of a simplified mathematical model to study the optimal orbit-to-orbit transfer. In fact, an Earth-comet SW1 mission is the typical scenario that can be approximated through a classical circle-to-circle orbit raising, in which the celestial bodies' (heliocentric) orbits are assumed as circular and coplanar. In that case, the spacecraft dynamics can be described with a (simplified) mathematical model which uses the classical spacecraft equations of motion in the polar (heliocentric) reference frame sketched in Figure 12. In this respect, the dynamics model and the optimization procedure are thoroughly described (and commented on) in a number of the authors' works [19; 47], and they are omitted here for the sake of brevity. Figure 11: Time variation of the components of the normal unit vector (in the RTN reference frame) during the optimal transfer with \(a_{c}=1\,\mathrm{mm/s^{2}}\) and an optical force model. Circle \(\rightarrow\) start; square \(\rightarrow\) arrival; black line \(\rightarrow\) radial component; red line \(\rightarrow\) tangential component; blue line \(\rightarrow\) normal component. Figure 10: Optimal transfer trajectory (black line) for an optical force model, when \(a_{c}=1\,\mathrm{mm/s^{2}}\). Black circle \(\rightarrow\) start; square \(\rightarrow\) arrival; star \(\rightarrow\) perihelion; orange circle \(\rightarrow\) Sun; blue line \(\rightarrow\) Earth’s orbit; red line \(\rightarrow\) comet SW1 orbit. In particular, the radius of the initial (or final) circular orbit is assumed as \(r_{0}\ =\ r_{\infty}\ =\ 1\) au (or \(r_{f}=6.0499\) au, i.e., a value that coincides with the semimajor axis of the comet SW1 orbit), while the initial polar angle is set to \(\theta(t_{0})=0\) for symmetry reasons; see Figure 12. Using such a two-dimensional approach to trajectory optimization, and considering the range of characteristic acceleration \(a_{c}\in[0.5,2]\) mm/s\({}^{2}\), the minimum flight time for an ideal or optical force model has been reassumed in Figure 13 as a function of \(a_{c}\). A comparison between the curves of Figure 13 and that of Figure 3 (which refer to a complete three-dimensional scenario), indicates that the results obtained through a two-dimensional approach approximate very well the values obtained with a more realistic model. This aspect is confirmed by Figure 14, which shows that the percentage difference between the minimum flight time obtained through a three-dimensional (\(\Delta t|_{\rm 3D}\)) and a two-dimensional (\(\Delta t|_{\rm 2D}\)) approach is below 4.6% for both the two force models. The solution of the optimal control problem also gives the value of the final polar angle \(\theta(t_{f})=\theta_{f}\), which can be used to evaluate the number of complete revolutions around the Sun during the transfer. In this case, the numerical results are reassumed in Figure 15, while Figure 16 shows a set of optimal transfer trajectories (in polar form), when an optical force model is considered in the simulations. Bearing in mind that \(\theta(t_{0})=0\), one observes that the trajectories of Figure 16 are consistent (and in some cases substantially coincident) with the trajectories reported in Figure 6. This is an important result, because it confirms a posteriori the usefulness of the two-dimensional approach, which requires a computation time of about two orders of magnitude less than that related to the three-dimensional model. Figure 12: Heliocentric polar reference frame and polar angle \(\theta\). Figure 14: Comparison between the minimum flight time obtained with a three-dimensional (\(\Delta t|_{\text{3D}}\)) and a two-dimensional (\(\Delta t|_{\text{2D}}\)) approach to trajectory optimization.