Suppose one has in-hand an accurate time-space trajectory in $\mathbb{R}^3$ of a (small) body,
say an asteroid or satellite—effectively a point.
To what extent does this trajectory determine the point masses that could 
gravitationally determine it (according to inverse-square gravitation)?
Is this highly underdetermined, in that there are many point-mass distributions
that would lead to the (exact) same trajectory, or does the trajectory essentially uniquely determine the masses?  Perhaps
this question only has a sharp answer with some assumptions on the size of the point masses, i.e.,
planetary or star-like, as opposed to spread-out asteroid belts or dust clouds...?
<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;![IOP][1]<br />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<sub>(Suggestive image from: "Spacetime symmetries and Kepler's third law,"
2012, *Class. Quantum Grav.*: 29. 217002 ([arXiv link][2])).</sub>


  [1]: http://ej.iop.org/images/0264-9381/29/21/217002/Full/cqg434450f1_online.jpg
  [2]: http://arxiv.org/abs/1202.2893