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...?

_{(Suggestive image from: "Spacetime symmetries and Kepler's third law,"
2012, Class. Quantum Grav.: 29. 217002 (arXiv link)).}

(

*Added 9Feb13*). In light of Ben Crowell's incisive analysis, and the various comments and answers (by Joel, Abhinav, Brendan, Theo) which point to fundamental nonuniqueness, permit me to rephrase the question:

Given an accurate (space-time) trajectory of a point-mass (planet) in a fixed coordinate system, and given a number $n$, under the assumption that there are $n$

point masses (stars) that might have caused the planet's trajectory solely due to inverse-square Newtonian gravitation, does the trajectory of the planet determine the fixed stars' masses and positions?stationary/fixed

It might be useful to distinguish general positions of the stars vs. special arrangements. If this can be answered for each $n$, then one could explore successively larger values of $n$.