Determination of the Optimal Jacobi Constant for Earth-Moon Transfers via L1 using the Theory of Functional Connections.

A detailed analysis of the mechanism for determining the Jacobi Constant (CJ ) in Earth-Moon transfers using the L1 Lagrangian point within the framework of the Circular Restricted Three-Body Problem (CRTBP) is presented. It is analytically demonstrated that CJ is veried via a Newton-Raphson algorithm that uses the sensitivity α = ∂CJ /∂(ε 2 ) = −26.2243, evaluated at L1. The characteristic values of CJ and their relationship with the amplitude parameter ε are identied. Unlike previous architectures, this work validates a complete round-trip mission with a total ∆V of only 78.9 m/s, a xenon consumption of 1.54 kg, and a duration of 16.3 days for a 250 kg spacecraft. This is possible thanks to a precise injection into the unstable L1 manifold (∆V = 42.6 m/s), completely passive lunar capture (∆V = 0 m/s), and manifold-assisted lunar escape (∆V = 18.5 m/s). The resulting mass eciency is 99.4%. Appendix A extends these results to the scaling for a real mission of 2869 kg (Tug L-RAIL), maintaining complete traceability from rst principles.