Universal Derivation of the Mass-Specific Source Invariant and the Critical Radial Lock: A Unified Trajectory Optimization Protocol for Lunar and Interplanetary Architectures

Abstract

This research paper formally introduces a novel astrodynamic framework designed to optimize fuel efficiency, orbital stability, and descent safety for Lunar Landers and next-generation space exploration. We derive the Mass-Specific Source Invariant, hereinafter referred to as the Source Invariant (Φ_i), a body-specific gravitational invariant representing the total potential capacity governed directly by the mass of a celestial body (Φ_i = G^2×m). Further, we introduce the Critical Radial Lock equation (r_l), a reverse-engineering mathematical tool that allows mission controllers to calculate the precise altitude required to achieve a specific target stability (S_target). We provide a rigorous dimensional analysis to validate the physical homogeneity of these equations. By applying this framework, we demonstrate that for the Moon, a stable Passive Orbit exists at an altitude of approximately 820 km (Zone II). At this Critical Radial Lock altitude, the gravitational signature reaches an optimized equilibrium (S = 0.50 ×10^(-10)), acting as a Safe Sanctuary with Minimum Fuel Consumption.