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I have a set of dates, position angles ($\theta$) and angular separations ($\rho$) for visual binary star. For example:

1994.3132  57.0    0.097
1996.1783  73.4    0.051
2006.1938 182.0    0.126
2009.2597 192.9    0.152
etc.

What are algorithms to calculate orbital elements and to plot the orbit of the binary star?

Are there available codes (or programs) for this calculations?

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Would scicomp.stackexchange.com be a better home for this question? –  Qmechanic Jan 8 at 22:46

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The good software for orbit calculation is "Binary" (Binary Star Combined Solution Package) by CHARA team:

http://www.chara.gsu.edu/~gudehus/binary.html

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If it's a binary, it's fairly simple (in comparison with a ternary system), because the stars of the binary orbit about their common barycenter in Kepler ellipses.

An orbit simulator, see here; replace the central star by the barycenter of the binaries.

Calculation of an Orbit from Three Observations. Kepler problem on Wikipedia.

If you don't find a ready-to-use software, try to solve it numerically by simulating the orbit of the binaries according to Kepler's law, and vary mass, distance, excentricity assumptions until they match to the observations. Use optimization methods, e.g. hill climbing algorithms, or gradient methods.

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Or, see for example: sourceforge.net/projects/phoebe –  Moriarty Jan 9 at 3:23
    
Needs some generalization to non-eclipsing binaries. –  Gerald Jan 9 at 4:09

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