I know that, if I wanted to map the circular orbit of an object by an iteration of time, I know that I can use the relationship between centripetal force and gravitational force...this is how we can find our orbital velocity.
But I also know that the second we move away from a circular orbit an into an elliptical one, I can't use the relationship between centripetal and gravitational forces to map out my path. Why not?
What would the force vectors look like when broken down into x and y vectors, where y is a line that runs through the object and planet (i.e. always perpendicular), and x is consistently tangent to the planet?