distinguishable orbits

I assigned Jagannath the following problem: What can you learn about the gravitational potential of a gravitating system in which you yourself lie (think the Milky Way) if all you have are streams of stars that trace out orbits, and all you can measure about those streams are their trajectories in angular coordinates (think RA, Dec). That is, what can you do if you get a snapshot of a few orbits, but you only get the angular shapes of those orbits? We went back and forth a bit between the answer everything and the answer nothing. I think it is the former, but of course it will depend strongly on the precision of the measurements! In some sense, all the answers are known; we are just on an intuition-building adventure.