That is a good and interesting thought.I suspect that after a period of time, the ball would become tidally locked to the sun and move along the track in synchrony with it.
I think the moon will have a stronger tidal force than the sun. And, the earth will definitely have a stonger tidal force than either.
I think what would happen is that the ball's spin rate would change so that (eventually) one side of the ball always faces the earth. After that the influence of the sun and moon become important and will gradually take energy away from the ball's kinetic energy. After that I'm not sure if it will lock to the sun or moon, or drift around.
This raises the question of whether tidal-friction is included with the statement of no-friction. Tidal friction would result from internal heating due to changing internal stresses/strains from the changing gravitational gradient inside the ball. Thingmaker would need to make a judgement here. His famous frictionless paint presumably would not impact the internal friction. But, since I didn't invent it, I can't say for sure.
If tidal friction is not allowed, then the ball would have no way to lose energy and become tidally influenced in any significant way. We can always say that slight changes in gravity will increase and decrease the ball's velocity by an unmeasurable amount. But this would be cyclic and there would be no steady decrease in energy.
This triggers another thought. Since the earth's surface moves at about 1000 miles per hour relative to a line draw from the earth's center to the sun's center. The ball could be rolled westward with that exact same velocity at noon, and the ball would be under the sun for a long time. In a sense, the ball would revolve around the sun as if the earth were not even there (EDIT: actually, I forgot about the moon, this would only apply if we didn't have a moon). However, eventually it will drift away from that state, and will likely be very relieved after baking under the equatorial sun for many centuries.
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