The question Is it possible to represent an orbit with just a periapsis position and 2 angular velocities? got me thinking, suppose I take a telescope and took 3 measurements of ISS angular position and their time stamps can I do OD? It occurred to me that 2 measurements are surely not enough if orbit is eccentric, then next question is, then does 3 measurement suffice? If so can it be done analytically ? If not then what could be an easy way to do numerically ?


As the answer to that question states:

In general, an epoch time and six scalar values that completely represent the state at the epoch time are needed.

The six values you can choose as you like, as long as they are independent variables. For example the position in x,y,z and the velocities in x,y,z directions would suffice. You can choose polar coordinates (r,$\theta$,$\phi$) instead.

With each of your measurements you get two angular positions. From the difference of them you can determine the angular velocities. This gives you already four values out of six. So far you don't have a measure of the distance of the observed object (r in polar coordinates). You can get this by doing another measurement, e.g. one orbit later. This gives you the period (trivial) and the orientation of the orbit (by some further calculations). So, you have six independent pieces of information, enough to exactly specify the orbit (in combination with your time measurement).

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    $\begingroup$ But, how is that three angular measurements preferably in same pass? $\endgroup$ – Prakhar Aug 1 '20 at 18:06
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    $\begingroup$ You can do that as well. The three measurements shouldn't be too close together: If the two resulting velocity vectors are almost collinear, the error in calculating the position / velocity in the third dimension will be huge. $\endgroup$ – asdfex Aug 1 '20 at 18:30
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    $\begingroup$ @asdfex Ceres was discovered on 1 January 1801. Piazzi observed Ceres a total of 24 times, the final time on 11 February 1801. To recover Ceres, Carl Friedrich Gauss, then 24 years old, developed an efficient method of orbit determination. Using the determined orbit, Ceres was rediscovered on 31 December 1801, von Zach and Heinrich W. M. Olbers found Ceres near the predicted position and thus recovered it. So observations over a month and a half were enough for an orbit period of 4.6 years. $\endgroup$ – Uwe Aug 1 '20 at 21:51
  • $\begingroup$ @Uwe - There are two factors that made it easier: Ceres is in a near circular orbit and Earth moved along its orbit as well so that they got quite a good parallax. I didn't find the results of Gauss' calculations, do you have them? $\endgroup$ – asdfex Aug 2 '20 at 9:34
  • $\begingroup$ @asdfex Maybe Gauß calculations of the orbit of Ceres were written as a personal letter to Zach or Olbers. There is a publication: "1809: Theoria Motus Corporum Coelestium in sectionibus conicis solem ambientium (Theorie der Bewegung der Himmelskörper, die die Sonne in Kegelschnitten umkreisen), Theory of the Motion of Heavenly Bodies Moving about the Sun in Conic Sections (English translation by C.H. Davis), reprinted 1963, Dover, New York." It may contain a part about Ceres. Gauß published in latin language. $\endgroup$ – Uwe Aug 2 '20 at 10:16

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