Animating Planetary Motion

This investigation provides formulae for animating planetary orbits including circular, elliptical, elliptical with precession, and sun centered elliptical. In each case, the calculation is continued to derive the appropriate force law. For the opposite approach, starting with a force law, and deriving the path, see for example Elliptic Orbit Solution based on initial conditions .

Elliptical orbit with no precession.

Some precession.

Some negative precession.

Large precession, where the precession period is a small multiple of the orbit period.

Large negative precession.

An implementation using the formulae is given here. An implementation using a numerical solution to the differential equation is given here. Here is a side-by-side comparison.

Formulae Differential Equation