Whenever two massive bodies interact via gravity it is useful to characterize the resulting motion. For example, is the situation more akin to scattering? Will an orbit result? If so, what shape? For both Newtonian gravity and general relativity circular orbits can result. In this section we explore the conditions for this to happen in both classical mechanics and general relativity.
Exercise Questions:
1. Initialize, but do not play (the Run button) the simulation. Now play the animation and note the period of the orbiter.
2. Calculate dφ/dτ for the orbiter using L/m .
3. Calculate dφ/dt for the orbiter.
4. Calculate the period of the orbiter from dφ/dt. Does it agree with the period shown in the animation?
5. If your answer for the period determined from Question 4 agrees with your answer to Question 1, most likely you have the correct dφ/dt for the orbiter. Calculate M/(dφ/dt)2. How does this number relate to the parameters of the orbit? Don't worry of you do not see this relationship, we will explore this further in the exercise, "t vs. r for Circular Orbits."
6. Now, click in the checkbox for the effective potential energy diagram, U(r). Sketch the actual effective potential energy diagram for the orbiter. Make sure to note the energy of the orbiter and the orbiter's initial position (as a circle) on the graph as well. Where does the circle's position appear on the graph and why? Do you see why this is the innermost stable circular orbit?
7. If you are careful, try resetting the animation and dragging the orbiter ever so slightly inward before starting the animation. What happens when you perturb the orbit?