# 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)?

-
Counterexample: you can come up with an ellipse or a parabola that passes through two points in the same times. –  Alyosha Mar 10 at 18:39

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.

-
However, if you move one of those two points by even the tiniest distance, the symmetry is broken and it narrows it down to exactly two orbits again, as in akhmeteli's answer. –  Nathaniel Mar 21 at 15:16
This will only be very approximate, though. In the nearly-collinear case the uncertainty in the orbit's orientation will be very large. –  Emilio Pisanty Mar 21 at 15:20
Ah, very good point, thank you. Would it be proper to accept this answer and create a new question, since it does answer the question I asked, but not the one I meant to ask (I'm interested in the 2-D case since this is for a computer game, not actual astronomy), or should I edit the original question? –  Peter Frauenglass Mar 21 at 19:01
@PeterFrauenglass: In the 2D case knowing the positions at apoapsis and periapsis still only narrows it down to two orbits, since this carries no information about the direction; in the 3D case this uncertainty is contained in the angular degeneracy. –  Chay Paterson Mar 25 at 17:31
@PeterFrauenglass: In general there will be an additional factor of two from not knowing the order in which the particle passes through the two points, so you can only ever narrow it down to four orbits: akhmeteli's plus their time reversals, but this is only one bit of information so it might not answer your question. A sensible way to fix the orbit in two dimensions is to get the position and velocity of the particle at one point in time, which is always sufficient if the gravitational sources are known. This is essentially your original idea but with an infinitesimal transit time. –  Chay Paterson Mar 25 at 17:40