## 3 Synthesis and Analysis of Planar SALT Trajectories
### Adaptation to More Complex Dynamical Models  
Patched SALT trajectories are easily adaptable to the BR4 BP model. When a single midcourse correction is allowed to be performed at the apogee of a SALT trajectory, the standard multiple-shooting procedure appears to work very well: a trajectory can be rapidly adapted by adjusting only the TLI impulse magnitude and phase and possibly  
Figure 14: Sample departing leg with a retrograde lunar flyby. The orbital angular momentum relative to the Moon in the rotating frame is negative; the spacecraft crosses the \(x\)-axis behind the Moon. adding a small apogee trajectory correction maneuver (TCM). The departure and arrival times (or, equivalently, the solar phase values at the departure/arrival epochs) are allowed to be fixed. Figure 16: Distribution of spacecraft polar angle values at the moment of TLI burn execution for SALT trajectories including a lunar flyby. Two distinct peaks at about 130 deg and 310 deg correspond to the 4 th and 2 nd quadrant trajectories with a direct flyby. The burn angle for retrograde flyby trajectories can be either close to 60 deg (2 nd quadrant trajectories) or about 240 deg (4 th quadrant trajectories). Figure 15: Initial conditions (the launch energy C\({}_{3}\) and the TLI burn phase) corresponding to the three types of SALT trajectories: without an intermediate flyby (blue), with a direct lunar flyby (light blue), or with a retrograde lunar flyby (red). Figure 17: Distribution of spacecraft polar angle values at the moment of TLI burn execution for SALT trajectories without an intermediate lunar flyby. Two peaks at about 160 deg and 340 deg correspond to the 4 th and 2 nd quadrant trajectories. The pattern is repeated almost exactly when shifting by 180 deg. Figure 18: Distribution of solar phase angles at the moment of TLI burn execution for SALT trajectories with a lunar flyby. Two distinct peaks at about 95 deg and 275 deg correspond to the 2 nd and 4 th quadrant trajectories with a direct flyby. For retrograde flyby trajectories, the pattern is not so clear, but small peaks around 140 deg and 350 deg values could be identified. Three examples of SALT trajectories before and after adaptation for each type of Sun-assisted transfers are depicted in Figures 21-23. In all the examples, the TLI burn magnitude changes by at most several m/s and its phase shift is in the order of a couple of degrees. The apogee correction maneuvers vary from 0.6 m/s for the second-type (direct flyby) trajectory to 34.5 m/s for the third-type (retrograde flyby) trajectory. Figure 19: Distribution of the total transfer duration for SALT trajectories without a lunar flyby. Figure 20: Distribution of the total transfer duration for SALT trajectories including a lunar flyby. Figure 21: A sample SALT trajectory of the first type (no flyby): the patched three-body approximation (thin) and upon adaptation to the BR4 BP (bold). The segments of the adapted trajectory corresponding to the departing, arriving, and exterior legs, are colored the same (red, green, and black, respectively). In the course of adaptation, the TLI impulse has increased from 3193 m/s to 3194 m/s. The burn phase has changed from 97.35 deg to 103.01 deg. The apogee TCM of 12.7 m/s has been added. Figure 22: A sample SALT trajectory of the second type (with a direct lunar flyby): the patched three-body approximation (thin) and upon adaptation to the BR4 BP (bold). As earlier, the adapted trajectory segments corresponding to the departing, arriving, and exterior legs, are colored the same (red, green, and black, respectively). Upon adaptation, the TLI burn has increased from 3163 m/s to 3166 m/s. The burn phase has changed from 221.84 deg to 221.43 deg. The apogee TCM of 0.6 m/s has been added. Converting a planar SALT trajectory to a three-dimensional transfer trajectory in a high-fidelity ephemeris model is a separate problem beyond the scope of this paper. However, the remark can be made that such a numerical procedure is almost always based on some sort of multiple-shooting method.