The Inconsistent use of $\omega$ in the RV Equation

Since the discovery of the first exoplanet orbiting a main-sequence star, astronomers have inferred the orbital properties of planets using stellar radial velocity (RV) measurements. For a star orbited by a single planet, the stellar orbit is a dilation and $180^\circ$ rotation of the planetary orbit. Thus, many of the Keplerian orbital properties of the star are identical to those of the planet. However, there is a notable exception: the argument of periastron, $\omega$, defined as the angle between the periapsis of an orbiting body and its ascending node. The argument of periastron of the star ($\omega_\star$) is $180^\circ$ offset from the argument of periastron of the planet ($\omega_p$). This distinction is important because some derivations of the RV equation use $\omega_\star$, while others use $\omega_p$. This discrepancy arises because commonly used derivations of the RV equation do not adhere to a single coordinate system. As a result, there are inconsistencies in the definitions of the Keplerian orbital parameters in various RV models, leading to values of the ascending node and $\omega$ that are $180^\circ$ offset. For instance, some packages, such as \texttt{RadVel} and \texttt{ExoFast}, report values for $\omega_{\star}$ that are identical to the $\omega_p$ values determined with other packages, such as \texttt{TTVFast} and \texttt{Orvara}, resulting in orbital solutions that differ by $180^\circ$. This discrepancy highlights the need for standardized conventions and definitions in RV modeling, particularly as we enter the era of combining RVs with astrometry.