Subject: Documentation of Mars Trajectory Optimization and 3BP Topology Energy Savings (1997–2002)

Claimant: William H. Clark II, PE (Texas License #74416, Inactive)

Professional and Academic Background

In 1997, at age 41, I was a licensed Professional Engineer in Austin, Texas, working as a consultant in the commercial construction industry with expertise in mechanical, electrical, and energy conservation engineering. (my company resume at the time) Over the preceding five years, I had published a dozen technical papers focused on dBase software I designed for engineering calculations. I operated an independent software startup, Bar X Software, which provided these calculation programs free of charge to readers of my two McGraw-Hill textbooks on electrical design and energy conservation.

My work in orbital frameworks began earlier; in 1982, I authored a manuscript proposing a new model of the solar system based on the Ten Body Problem of celestial mechanics (9 planets + Earth’s moon). A formal review letter from NASA during that period stated:

“Your work was reviewed by both an engineer and a physicist… They both agreed that your  concepts about the Unified Field Theory are so profound as to be years ahead of the present thinking in the areas in which you delved.”

In the spring of 1997, following correspondence with Dr. Victor Szebehely at the University of Texas at Austin regarding my solar system model, I enrolled in his graduate course, ASE 388P (Regularization). I presented my solar system model as my class project. Dr. Szebehely loved my work.

"All you need to do is add time to your model and you’ll get your name lit up in lights.”

Dr. Szebehely later wrote a formal letter of recommendation for my admission to the graduate aerospace engineering program at Texas. I began graduate classes in the fall of 1997. Professor Szebehely passed away one month later. My advisor thereafter was Dr. Roger Broucke.

Algorithmic Development and Technical Breakthrough

In the spring of 1998, following the high-profile failure of several Mars missions, I obtained approval from Dr. Broucke to develop a computer-based optimization model for Mars trajectories. Dr. Broucke provided a copy of his very compact, efficient Runge-Kutta 7/8 integrator to use in my Mars model.

This Mars model was foundational to my solar system model, and proved several important aspects.

The Mars algorithm was structured as the standard five-thrust NASA mission:

The spacecraft departs Earth in the ecliptic plane on a hyperbolic escape trajectory.

The state vector is evaluated at the Earth’s Sphere of Influence (SOI).

The spacecraft transits toward Mars.

At conjunction, the spacecraft transitions to the orbital plane of Mars.

The trajectory continues to the Martian SOI, where the final approach is optimized via a two-body approximation, intended as a baseline prior to a full 7/8 numerical integration into a parking orbit.

Initially, standard nonlinear optimization routines, including software provided by my graduate advisor Dr. David G. Hull, failed to converge on a solution. During a consultation with Dr. Broucke, we realized that because all the forces in my model were gravitational, the system could be integrated backward from the point of conjunction. (Gravity is a central force – this is how astronomers study the evolution of our solar system, by integrating backward.)

By restructuring my algorithm to utilize backward integration, I achieved rapid convergence. The solution eliminated the requirement for a complex nonlinear optimization routine, executing 26 pages of compiled Fortran code in approximately 30 seconds on a 333 MHz PC. I verified the algorithm by benchmarking it against a standard Hohmann Transfer (adapted for non-circular, non-coplanar orbits); the resulting numerical outputs matched perfectly.

Integration of Three-Body Topology and Fuel Optimization

I subsequently hypothesized that the termination point at the Martian SOI corresponded closely with the L1 Lagrange point of the Sun-Mars Three-Body Problem (3BP). To exploit this 3BP topography, I adjusted the algorithm to target the L1 point exactly where it intersects the Sun-Mars free-return trajectory—the characteristic figure-eight-shaped loop between Mars and the Sun.

My optimization methodology functioned through the following systematic steps:

Targeting the L1 Intersection : The model stopped at Mars’ SOI at the L1 Lagrange point.

Topological Boundary Perturbations : From this initial state vector, I took small, incremental "baby steps" along the geometry of the SOI.

Orbital Convergence Verification : For each iteration, the algorithm calculated the required insertion energy needed to capture into the identical, fixed Martian parking orbit using the Rung Kutta 7/8 integrator.

Energy Mapping : Initially, as the starting locations shifted along the SOI, the total insertion energy remained completely unchanged.

Discovery of the Topological Drop : At...