## 3 Synthesis and Analysis of Planar SALT Trajectories
### Classification and Features of Patched SALT Trajectories  
Let us classify and characterize the patched SALT trajectories we obtained. They all can be grouped into three categories, depending on the existence and type of an intermediate lunar flyby on the departing leg. Sample trajectories of each type are shown in Figure 12. The first type is comprised of SALT trajectories not approaching the Moon closer than 60,000 km on the departing leg (no flyby). The other two types of trajectories (which amount to 14% of the whole database of patched SALT trajectories) include a lunar flyby. For a second-type SALT trajectory (see Figure 13), the spacecraft crosses the \(x\)-axis in front of the Moon (when observing from the Earth) and has a positive orbital angular momentum \(w_{z}\) in the rotating frame at the moment of flyby. We will thus further refer to such a flyby as _direct_ (note that in [31; 33] it is called a _trailing side flyby_). On the contrary, a SALT trajectory of the third type (Figure 14) includes a _retrograde flyby_ (a _leading side flyby_  
Figure 11: Earth–Moon RoP re-entry conditions corresponding to the exterior legs successfully patched with a departing leg. The color reflects the Jacobi integral value at re-entry. in [31; 33])--the spacecraft crosses the \(x\)-axis behind the Moon and has a negative orbital angular momentum. As can be readily deduced from Equation (A12) of Appendix A.2, the orbital angular momentum in the rotating frame \(w_{z}\) is related to its inertial-frame counterpart \(W_{z}\) at an arbitrary moment by the simple expression \(W_{z}=w_{z}+r_{2}^{2}\). Particularly, at the moment of flyby, we have \(W_{z}=w_{z}+r_{\pi}^{2}\). If the flyby distance \(r_{\pi}\ll 1\), then \(W_{z}\approx w_{z}\). It is convenient that the initial conditions (the launch energy \(C_{3}\) and the TLI burn phase) for the patched SALT trajectories almost do not depend on a specific low-Earth parking orbit and can be visualized in a single chart (Figure 15). Excluding transfers with an extra revolution around the Earth before an intermediate lunar flyby, one can conclude  
Figure 12: Three types of SALT trajectories: (I) without any intermediate lunar flyby, (II) with a direct lunar flyby, (III) with a retrograde lunar flyby. The Moon’s orbit is marked by the dashed gray line. Figure 13: Sample departing leg with a direct lunar flyby. The orbital angular momentum with respect to the Moon in the rotating frame is positive; the spacecraft crosses the \(x\)-axis in front of the Moon. that SALT trajectories of the second or third type require executing the TLI maneuver at a point in the parking orbit in which phase \(\theta_{0}\) is in the range of \(215\ldots 295\) deg. The lower part of this range (\(215\ldots 225\) deg) corresponds to second-type trajectories, while the TLI burn phase of third-type trajectories is bounded between 235 deg and 295 deg. As expected, no-flyby trajectories do not exhibit any pattern in \(\theta_{0}\) and require a launch energy of at least \(-0.8\) km\({}^{2}\)/s\({}^{2}\) (the TLI impulse of about 3.19 km/s in the 200 km parking orbit). Including an intermediate flyby allows one to reduce the launch energy as low as up to \(-2.1\) km\({}^{2}\)/s\({}^{2}\). For all three types of SALT trajectories, the initial Sun-Earth-spacecraft geometry is predictably crucial. It is expressed the most in terms of the TLI burn polar angle \(\alpha_{0}\) relative to the Sun-Earth line. The majority of trajectories with a direct lunar flyby demand this angle to be approximately either 130 deg or 310 deg (see Figure 16). The former case corresponds to fourth quadrant trajectories, whereas the latter is for second quadrant trajectories. The burn angle for trajectories with a retrograde flyby is about 60 deg (for second quadrant trajectories) or close to 240 deg (for fourth quadrant trajectories).