I am doing an IA for an IB math course and my topic is "have/will all the planets ever come within a certain degree of each other?" I am starting out by nullifying certain aspects that would make deriving an equation for each incredibly hard, for example the perturbations due to gravity on orbital eccentricity (so no gravitational law) and ignoring the slight deviations in the plane that the planets lie on (x and y only, not z). Does anyone have a good link of the planet's orbital equations or how to derive a simple formula that is just an ellipse in cartesian form (x,y plots, no z)that takes into account time?
One aspect that I must stress is that I will be doing this in 2D (x and y), not taking into account z. My graph will have the perspective that you are looking down on our solar system, without depth.
From there I will program each of the orbitals and find an angle (probably that of the diameter of Jupiter) to constrain the "intersection". Finally, I will code a system that runs until 4 or more planets come into the same angle as each other, and run this very far into the future/past.