# Solar Sail Transfer Trajectory Design for Comet 29P/Schwassmann-Wachmann 1 Rendezvous
## 2 Mathematical Preliminaries and Models  
The problem addressed in this section is to analyze the rapid transfer trajectory, in a three-dimensional heliocentric mission scenario, of a spacecraft whose primary propulsion system is a classical (photonic) solar sail of assigned characteristics. In this context, assume that the solar sail-based spacecraft initially (i.e., at time \(t=t_{0}\triangleq 0\)) covers a Keplerian (three-dimensional) orbit which coincides with the Earth's orbit around the Sun, whose gravitational parameter is \(\mu_{\infty}\). As is well known, such a specific scenario is consistent with a solar sail deployment just outside the Earth's sphere of influence [21], when the spacecraft leaves the planet using a Keplerian parabolic escape trajectory.  
In an orbit-to-orbit transfer, the spacecraft uses the propulsive acceleration given by the solar sail propulsion system to reach, at time \(t=t_{f}>t_{0}\), a Keplerian orbit which coincides with the SW1 heliocentric trajectory. According to the common assumptions used in the preliminary phase of the trajectory design, during the transfer, that is, during the time interval \(\Delta t\triangleq t_{f}-t_{0}\equiv t_{f}\), the spacecraft is subjected to the Sun's gravitational acceleration and, possibly, solar sail-induced thrust. Moreover, in an orbit-to-orbit transfer the initial (or final) angular position of the spacecraft along the parking (or target) orbit is left free, so that the initial (or final) spacecraft true anomaly \(\
u_{0}\triangleq v(t_{0})\) (or \(\
u_{f}\triangleq v(t_{f})\)) is obtained as an output of a trajectory optimization process that extremizes (i.e., optimizes) a given performance index. On the other hand, the remaining classical orbital parameters that define the parking (i.e., the Earth) and the target (i.e., the comet SW1) orbit are considered assigned, and they are reassumed in Table 1. In particular, this table gives the value of the semimajor axis \(a\), eccentricity \(e\), orbital inclination \(i\), argument of perihelion \(\omega\), and right ascension of the ascending node \(\Omega\), which are obtained through the well known JPL Horizons system.  
Note that the comet SW1 orbital elements reassumed in Table 1 are evaluated at the beginning of August 2023. In fact, as discussed by [PERSON] et al. [22], the heliocentric orbit of comet SW1 is (strongly) perturbed by Jupiter's gravitational attraction, and the simulations show that the Jupiter conjunction of year 2038 will (roughly) double the eccentricity of the comet orbit.  
In a solar sail-based mission scenario, the time variation of the thrust vector components is typically calculated by minimizing the total flight time [23], that is, by studying the (rapid) transfer trajectory which maximizes the performance index \(J\) defined as  
\[J\triangleq-\Delta t\equiv-t_{f}. \tag{1}\]  
The procedure used in this work to maximize the performance index \(J\) and, then, to obtain the minimum flight time related to the rapid transfer trajectory, has already been validated in a number of different mission scenarios which involve the solar sail-based spacecraft transfer towards Solar System small bodies, as also illustrated in some recent authors' works [24; 25]. An interesting approach to the solar sail-based trajectory optimization, in missions toward Solar System small bodies, is also discussed in the works by [PERSON] et al. [26] and [PERSON] et al. [27]. In particular, the procedure used in this work will be (briefly) illustrated at the end of this section, for the sake of completeness. The next subsection, instead, illustrates the mathematical model used to describe the solar sail-based spacecraft dynamics in the three-dimensional heliocentric scenario.