# Solar Sail Transfer Trajectory Design for Comet 29P/Schwassmann-Wachmann 1 Rendezvous
## 3 Numerical Simulations  
The optimization procedure discussed in the previous section has been applied to evaluate the rapid Earth-comet SW1 transfer as a function of the spacecraft characteristic acceleration value in the range \(a_{c}\in[0.5,\,2]\,\mathrm{mm}/\mathrm{s}^{2}\), with a step of \(0.1\,\mathrm{mm}/\mathrm{s}^{2}\); see the red circles in Figure 3 for example. Bearing in mind the current solar sail technology level [16; 23], the range below the (usually considered as canonical [34]) value of \(1\,\mathrm{mm}/\mathrm{s}^{2}\) is consistent with a medium-high performance solar sail, which hopefully will be available in the near future. On the other hand, the upper part of the selected range of \(a_{c}\) models the behavior of a (more futuristic) high performance sail [41; 42; 43; 44], which uses advanced materials [45; 46] to improve the propulsive performance. In order to evaluate the impact on the transfer performance of the sail thrust characterization, the orbit-to-orbit transfer has been optimized by considering both the ideal and the optical force model, using the normalized coefficients \(\{b_{1},\,b_{2},\,b_{3}\}\) reassumed in Table 1. The results, in terms of optimal flight time \(\Delta t\) as a function of \(a_{c}\), are given by Figure 3 for the two force models. Figure 3 clearly shows that the (optimal) flight time of a medium-high performance solar sail is well above 10 years, with a value of \(\Delta t\simeq 25\,\mathrm{years}\) when \(a_{c}=0.5\,\mathrm{mm}/\mathrm{s}^{2}\) for both the force models considered in the numerical simulations. On the other hand, an orbit-to-orbit transfer time below 10 years can be reached only through a high performance solar sail. In this context, for example, a sail with \(a_{c}>1.8\,\mathrm{mm}/\mathrm{s}^{2}\) is required when an optical force model is used to describe the thrust vector. This aspect is due to the combination of a high value of the semimajor axis and a low eccentricity of the comet SW1 heliocentric orbit, which gives a perihelion (or aphelion) distance of about 5.78 au (or 6.32 au). In fact, a classical solar sail is quite ineffective at such distances from the Sun, because the maximum magnitude of the solar sail propulsive acceleration \(\max\|\mathbf{a}\|\) varies as \(1/r^{2}\), as described by Equation (14). In this specific mission scenario one obtains that the dimensionless ratio  
Figure 3: Minimum flight time \(\Delta t\) as a function of the spacecraft characteristic acceleration \(a_{c}\in[0.5,\,2]\,\mathrm{mm}/\mathrm{s}^{2}\), in an Earth–comet SW1 orbit-to-orbit transfer using an ideal or optical force model. \(\max\lVert\mathbf{a}\rVert/a_{c}\) is in the range [0.025, 0.03] when the spacecraft completes the rendezvous, as illustrated by the curve of Figure 4, which describes the variation of the ratio \(\max\lVert\mathbf{a}\rVert/a_{c}\) as a function of the spacecraft (final) true anomaly \(\
u_{f}\) along the comet orbit. In other terms, at the end of the mission the solar sail-induced propulsive acceleration magnitude is only a small fraction (i.e., a value below 3%) of the initial maximum value, which roughly coincides with \(a_{c}\) by construction. The numerical simulations show that the spacecraft with a medium-high performance solar sail completes several revolutions during the optimal orbit-to-orbit transfer. This point is highlighted by Figure 5, which shows the variation in true longitude \(\Delta L\) during the transfer. Note that the integer part of the ratio \(\Delta L/(2\pi\,\text{rad})\) indicates the revolutions around the Sun completed by the spacecraft before reaching the comet SW1 heliocentric orbit. The value of \(\Delta L\) is consistent with the curves reassumed in Figure 6, which shows the ecliptic projection of the optimal transfer trajectory for some values of \(a_{c}\), using an optical force model. Figure 4: Variation of the dimensionless ratio \(\max\lVert\mathbf{a}\rVert/a_{c}\) as a function of the spacecraft (final) true anomaly \(\
u_{f}\).