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The radial velocity technique has apparently been used to discover exoplanets with extremely long orbital periods of ~10,000 days. For example, see this link:

https://exoplanetarchive.ipac.caltech.edu/cgi-bin/TblView/nph-tblView?app=ExoTbls&config=planets

If the period of a planet is ~10,000 days, then the period of the star must be the same. However, this would mean that observations would need to be made over ~30 years to get a complete picture of the radial velocity variation of the star, a time which is far too long. How then are radial velocity measurements used to discover such planets?

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You don't need to collect data for one cycle in order to determine the period.

From Newtonian's physics, we know precisely how a trajectory of one particle orbitting around another one would be. Since the instantaneous velocity (v_inst) at any point on the trajectory is tangential to the point on the trajectory, we can decompose the v_inst into the radial velocity which we can match with the observation from, e.g., redshift. With assumption of constant v_inst and trajectory, we can determine the period from just a couple of observations (depending on how accurate the redshift can be measured, and the number of variables in the equation).

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  • $\begingroup$ You need more than 2 measurements to estimate the parameters of an elliptical orbit. $\endgroup$
    – ProfRob
    Commented Dec 9, 2018 at 8:45
  • $\begingroup$ @RobJeffries How many measurements are required to extrapolate the parameters of an elliptical orbit? $\endgroup$
    – BenjaminF
    Commented Dec 9, 2018 at 15:17
  • $\begingroup$ I would say, as with any data, that the uncertainty decreases with more measurement. In this case it seems to me that we are reconstructing a sinusoidal curve. $\endgroup$
    – Jack R. Woods
    Commented Feb 12, 2019 at 16:58

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