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I think you are referring to the orbits of visual binaries, since you are asking about parameters measured from an ellipse. If the orbital plane were at right angles to the line of sight, then the focus of the ellipse defined by the motion of the secondary star would be coincident with the primary star. When you change the inclination, this is not the case, ...

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Bond et al. (2017) measure the orbital period of the Sirius system to be $50.1284 \pm 0.0043$ years. I believe this is the most precise and accurate value (I cannot find any more recent papers, with new determinations, that cite this paper). An earlier, comprehensive study by Gatewood & Gatewood (1978) gave $50.090\pm 0.056$ years; consistent with the ...

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Calculations like this one for the orbital period of a binary system are hardly ever done analytically. Astronomers usually take some known approximations and extend it to system like Sirius, and get an estimate for their period numerically (and a good one, by the looks of it). If you're curious, we can derive this version of the 3rd law by considering the ...

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