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...?
        IOP http://ej.iop.org/images/0264-9381/29/21/217002/Full/cqg434450f1_online.jpg
        _{(Suggestive image from: "Spacetime symmetries and Kepler's third law," 2012, Class. Quantum Grav.: 29. 217002 (arXiv link)).}