Here are some animated gif movies relevant to GR. I have calculated these by numerical integration of the equations of motion. See the scripts for details. The black circles indicate the Schwarzschild event horizon.
- Stable circular orbit. Just larger than last stable orbit.
- Unstable circular orbit. Just smaller than last stable orbit.
- Apsidal precession.
- Apsidal precession, ticking off the turning points.
- Time travel on the cheap: this orbit allows you to slow down your time at no cost in energy.
Here are some related images showing paths of photons launched from initial radii scaled in terms the the gravitational radius, GM/c2: