Can an orbit be calculated using two points and transit time? Working in only two dimensions and assuming that the central body is at the origin of the coordinate system, given two points in space and knowing the transit time between those points, as well as the direction of motion, is it possible to calculate a body's orbit?
It seems to me that there should be enough information - at $T=0$, the object was at $(θ_0, r_0)$ and at $T = T_1$, $(θ_1, r_1)$.
Two points + knowing one focus gives a set of ellipses - but the additional information of transit time seems to me like it should be enough to narrow it down to 1-2 orbits in the general case, travelling in opposite directions if there are two. Is it possible? If so, how would I go about doing so (numeric approximations are fine)?
 A: In general, two points and transit time do not define a single orbit. Let us imagine an (non-circular) ellipse with a focus in the origin and the major axis on the abscissa. If we build a second ellipse by reflecting the first ellipse symmetrically with respect to the ordinate, these two ellipses will intersect in two points (A and B) on the ordinate, and the origin will be a focus of each of the two ellipses. So a point can travel from A to B either along one of the ellipses or along the other one. The transit time will be the same for both ellipses (if we choose the right arcs).
EDIT(12/22/2012): I don't think I'll have time to give a complete answer to the edited question, but I would suggest the following approach. The equation of an ellipse with a focus in origin O has three free parameters in polar coordinates: major semi-axis, eccentricity, and the direction of the major semi-axis. If the points in question are A and B, distances OA and OB and angle AOB should fix these three parameters, yielding (in a general case) a discrete set of solutions. The equation for the transit time can be obtained by finding the area of ellipse sector AOB (integration in polar coordinates seems to yield an elementary function) and using the second and the third Kepler laws.
A: In general the answer is no.
In addition to akhmeteli's counterexample above, there is a more dramatic case where an observation fails to narrow down the number of possible orbits even to a finite set: if the two observations correspond to the planet's positions at periapsis and apoapsis, we will have gained information about the semimajor axis and eccentricity, but cannot fix the orientation of the orbit; this is essentially because we don't get any information about the longitude of the ascending node.
You can see this by rotating the orbit around the axis that runs through apoapsis, periapsis, and the parent star. There are continuously many possible orbits.
A: Have you looked at Lambert's Problem?
It can be used to solve for a conic orbit that goes from pointA at timeA to pointB at timeB around the same centre of attraction. It only works for the two body problem though, so it wouldn't take into account the gravity of any other bodies.
It is used in preliminary calculations for interplanetary missions.
