Here is the question I’m answering.
It’s a trick question! No possible height profile will perfectly reproduce Kepler orbits. The problem is that in the solar system, any given planet moves in two dimensions around the sun. But since the bowl is a curved surface, the balls wobble up and down through three dimensions, and you can’t match these different scenarios up perfectly.
The dynamics of a planet orbiting the sun come out of the Newtonian gravitational potential
So you might think that if you just make the height of the bowl inverse proportional to the distance from the center, so that , the balls would follow Kepler orbits. After all, their potential energy would be the same as the potential energy of a planet orbiting the sun, right?
We need to look more closely at the variable . For the case of a planet around the sun, is simply the distance from the planet to the sun. But for the balls circling the bowl, there are two possible interpretations of . One interpretation is to take a string, lay it flat on the bowl, and measure the distance along the bowl to the center. That would be . The problem with this approach is that the space is curved. If you were to measure the ratio of the circumference of a circle to its radius using defined this way, you would not get , You would get something that depends on . How could you then reproduce orbits through flat Newtonian space?
Instead of treating the surface of the bowl as a two-dimensional space, you might try to treat its projection as a two-dimensional space. So get up directly above the exhibit and look straight down at it with one eye closed. Then you’re looking at a flat space, so could you reproduce Kepler orbits there?
No, because the projection treats the radial and angular directions differently. If a ball has a true velocity of and is going around the center of the bowl in a circle, then in projection it still has an apparent velocity of . On the other hand, if the same ball were plunging straight in towards the center, its velocity would appear slower by a factor of the slope of the bowl, because you wouldn’t notice the portion of its velocity that was up/down in real 3D space. The angle at which a ball appeared to be moving would be distorted by this effect.
If you designed the bowl so that the period of circular orbits followed Kepler’s third law, then in general the projections of balls wouldn’t follow conic sections any more. Projected angular momentum would not be conserved because real angular momentum is conserved, and the projection would hide different proportions of that at different times.
So, while the Kepler exhibit is cool to look at, as best I can tell you can’t truly make it mimic the orbits of planets around the sun.