Can you show us the solution to the Three Body Problem that this is based on?
Straw-man fallacy - you're attacking someone who is not part of the conversation. We cannot show you the solution to the Three Body Problem that this is based on, because we are not privy to that information. I can however provide you with many case solutions to 3-Body Problems and well as a method to
mostly solve the Sun-Earth-Moon problem, it's only failure being the inability to provide perfect values into the infinite future.
Known solutions to 3-Body Problems:
In 1767, Leonhard Euler found three families of periodic solutions in which the three masses are collinear at each instant.
In 1772, Lagrange found a family of solutions in which the three masses form an equilateral triangle at each instant.
In work summarized in 1892–1899, Henri Poincaré established the existence of an infinite number of periodic solutions to the restricted three-body problem.
In 1911, William Duncan MacMillan found one special solution and in 1961, Sitnikov improved this solution, to the Sitnikov problem.
In 1967 Victor Szebehely and coworkers established eventual escape for the Pythagorean three-body problem using numerical integration, while at the same time finding a nearby periodic solution.
In the 1970s, Michel Hénon and Roger A. Broucke each found a set of solutions that form part of the same family of solutions: the Broucke–Henon–Hadjidemetriou family.
In 1993, a solution with three equal masses moving around a figure-eight shape was discovered by physicist Cris Moore.
In 2013, physicists Milovan Šuvakov and Veljko Dmitrašinović at the Institute of Physics in Belgrade discovered 13 new families of solutions for the equal-mass zero-angular-momentum three-body problem.
The general case of the three-body problem does not have a known solution and is addressed by numerical analysis approximations.
In many cases such a system can be factorized, considering the movement of the complex system (planet and satellite) around a star as a single particle; then, considering the movement of the satellite around the planet, neglecting the movement around the star. In this case, the problem is simplified to two instances of the two-body problem. The effect of the star on the movement of the satellite around the planet can then be considered as a perturbation. The general statement for this three-body problem is as follows.
At an instant in time, for vector positions xi and masses mi, three coupled second-order differential equations:
x ¨ 1 = − G m 2( x 1 − x 2/ | x 1 − x 2 | 3 )− G m 3( x 1 − x 3/ | x 1 − x 3 | 3)
x ¨ 2 = − G m 3 (x 2 − x 3 /| x 2 − x 3 | 3 )− G m 1 (x 2 − x 1 /| x 2 − x 1 | 3)
x ¨ 3 = − G m 1 (x 3 − x 1 /| x 3 − x 1 | 3 )− G m 2 (x 3 − x 2 /| x 3 − x 2 | 3)
The time evolution of the system is believed to be chaotic. The use of computers, however, makes solutions of arbitrarily high accuracy over a finite time span possible using numerical methods for integration of the trajectories.
[reference: Wikipedia]