How are exoplanets with 10,000 day periods discovered with the radial velocity method? 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&amp;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?
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).