Back at my notes over the next few weeks, I could not get the reverse integration out of my brain. It’s like those grocery store jingles that keep on rattling around your head… Then –
Boom!
My code was already set up in two halves, as two Three Body Problems, divided at conjunction. Conjunction is where the Earth, Sun, and Mars line up. Earth and Mars are in different orbital planes, but the planes intersect at conjunction, so I had a thrust there in my trajectory to move the spacecraft into Mars’ orbital plane.
The fixed endpoints of my model were parking orbits around Earth and Mars, as shown. I realized I could use the same method to reach Earth – IF I INTEGRATED BACKWARD FROM CONJUNCTION TO EARTH. Since my model only had gravity forces from the Sun, Earth, and Mars; this was okay, because gravity is a central force.
Since my model did not have solar radiation or any other nonlinear forces, the reverse integration would be EXACT, and I was not limited to the baby steps in traditional/standard nonlinear optimization solvers. Here is what the “final approach” to Earth looks like going backward in TIME. (The exact same technique was used for the final approach to Mars going in forward TIME.)
Here is the flow chart diagram for the Earth part of the code. It’s called HAWKEYE and internal subroutines INITIAL, TRAPE, and TAILHOOK (after the tailhook used by jets to land on aircraft carriers, catching a trip wire to stop).
By dividing an extremely nonlinear problem into two equal problems, they could each be solved independently, then added together at conjunction to create a seamless flight path from Earth to Mars.
I had planned to plug this program into a nonlinear optimization monster to generate a final solution that saved even more energy, but never got around to that part. Nobody was the least bit impressed with my claim to have reached Mars without a nonlinear optimization routine. I suppose they assumed I had cheated by setting up a much simpler problem.
I insisted it was a full 3D simulation with the exact 1960 ephemerides, but all I got was.
No feedback until you pass the qualifying exams.
HELLO?! This is my MASTERS Thesis. M.A.S.T.E.R.S.
No response.