## 3 Synthesis and Analysis of Planar SALT Trajectories
### The \(L_{2}\) Lunar Gateway  
It is convenient to design a SALT trajectory in reverse order, starting with the arriving leg. At this phase of flight, exterior ballistic capture happens: the spacecraft \"dives\" into the \(L_{2}\) neck of the zero-velocity surface. In the planar Earth-Moon CR3 BP, the integral manifold \(\mathcal{J}(\widetilde{J})\) for low-energy values of \(\widetilde{J}\) (that is, \(3<J_{L3}\leq\widetilde{J}\leq J_{L2}\); see also Appendix A.1 about the critical values of the Jacobi integral) is divided into two non-overlapping parts by the stable invariant manifold of the planar Lyapunov orbit with the same value of \(\widetilde{J}\)[3; 36], and ballistic capture can occur in a planar SALT trajectory only if its arriving leg belongs to the interior of the corresponding two-dimensional stable manifold tube (Figure 6). In the \((x,\dot{x})\) plane, manifold trajectories, when crossing the Earth-Moon RoP boundary, form a closed curve limiting _the \(L_{2}\) lunar gateway_ (Figure 7). Let \(G\) denote a set of inner points of the gateway. For any point \([x_{P},\dot{x}_{P}]\) of the gateway, the corresponding coordinate \(y_{P}\) may be found from the condition of belonging to the RoP boundary; after that, \(\dot{y}_{P}\) is determined by the energy constraint \(J(x_{P},y_{P},\dot{x}_{P},\dot{y}_{P})=\widetilde{J}\). The closer \(\widetilde{J}\) to \(J_{L2}\), the smaller the lunar gateway, shrinking to a single point at \(J=J_{L2}\approx 3.184\) (see Figure 8). Thus, inserting the spacecraft into any circulman orbit with \(J>3.184\) is impossible without additionally performing a lunar orbit insertion maneuver. The numerical propagation of any initial condition \([x_{P},y_{P},\dot{x}_{P},\dot{y}_{P}]\) in the CR3 BP model generates a ballistic capture trajectory (Figure 6) passing by the Moon at a selenocentric distance \(r_{\pi}\), with the argument of perilune \(\omega_{\pi}\) (in the planar case, it stands for the angle between the \(Cx\) axis of the Earth-Moon rotating reference frame and the Moon-perilune line). The points corresponding to a specified value of \(r_{\pi}\) can be grouped in a single gradient-colored contour line. Figure 9 shows such a line for \(r_{\pi}=3141\) km on the \(L_{2}\) lunar gateway with \(J=3.06\). The color along the line indicates the \(\omega_{\pi}\) value for the associated ballistic capture trajectory. If we want the arriving leg to have specific values of \(r_{\pi}\) and \(\omega_{\pi}\), the corresponding gateway point should be targeted. If the Jacobi integral value of the  
Figure 5: Earth–Moon and Sun–barycenter rotating reference frames \(Cxy\) and \(Cx^{\prime}y^{\prime}\). The radius vector of the spacecraft in the Sun–barycenter frame is described by the polar angle \(\alpha\). desired (nominal) lunar orbit is greater than \(J_{L2}\), the spacecraft has to enter a gateway with \(J\lesssim J_{L2}\) and then perform an LOI burn. Figure 6: Sample ballistic capture trajectory with the closest lunar approach of 1403 km (\(r_{\pi}=3141\) km) at a point with \(\omega_{\pi}=83.5\) deg. The Jacobi integral value equals \(J=3.06\) along the capture trajectory. Figure 7: \(L_{2}\) lunar gateway \(G\) in the \((x,\dot{x})\) plane. The region is formed by stable manifold trajectories of the \(J=3.06\) planar Lyapunov orbit when crossing the Earth–Moon RoP boundary. To estimate a perilune impulse required to insert the spacecraft in a nominal lunar orbit with \(J>J_{L2}\), one can use the approximate expression \(J\approx 3\left(1-\mu\right)+2W_{z}-2E\) for the Jacobi integral via the osculating elements in the Moon-centered inertial frame (see Appendix A.2). Here, \(W_{z}\) is the \(z\)-component of the orbital angular momentum and \(E\) is the Keplerian energy of the spacecraft's capture orbit around the Moon.