Universal Derivation of 'Deepak Source Constant' (D_s) and 'Deepak Radial Lock' (r_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 "Deepak Source Constant" (D_s), a body-specific gravitational invariant representing the Total Potential Capacity of a celestial body (D_s = G^2.m). Furthermore, we introduce the "Deepak Radial Lock" equation (r_lock), 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 Radial Lock altitude, the gravitational signature reaches an optimized equilibrium (S = 0.50 x 10^(-10)), acting as a "Safe Sanctuary" with Minimum Fuel Consumption.