TITLE: To what extent does trajectory determine gravity sources?
QUESTION [11 upvotes]: 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...?

        
(source: iop.org) 
        (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 $n$ stationary/fixed 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?

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 $n$.

REPLY [2 votes]: Although this question has been answered many times, I would like to add the following simple way to generate examples.  It is clear that in the plane any central force, no matter how complicated,  generates circular orbits as special cases.  There are many ways to distribute matter in three space to generate such a force field.  For example, any distribution of mass (which can, of course, consist of point masses) along the $z$-axis
will produce such a field in the $x,y$-plane.  As a side remark, such configurations occur
in practice---e.g., as the gravitational field generated by a wire.
I might add for those interested in the theme of trajectories that there is a considerable body of work on the question of which families of curves can occur as the trajectories of force fields, including a monograph
"Differential-geometric aspects of dynamics",  by Edward Kasner and his associates, much of which is easly accessible online. This is, however, not directly relevant to the query.