Addressing Systematics in the Traceback Age of the $\beta$ Pictoris Moving Group

We characterize the impact of several sources of systematic errors on the computation of the traceback age of the $\beta$ Pictoris Moving Group ($\beta$PMG). We find that uncorrected gravitational redshift and convective blueshift bias absolute radial velocity measurements by $\sim$ 0.6 kms${}^{-1}$, which leads to erroneously younger traceback ages by $\sim$ 2 Myr. Random errors on parallax, proper motion, and radial velocity measurements lead to an additional bias of $\sim$ 1.5 Myr on traceback ages. Contamination of astrometric and kinematic data by kinematic outliers and unresolved multiple systems in the full input sample of 76 members and candidates of $\beta$PMG also erroneously lowers traceback ages by ${\sim}$ 3 Myr. We apply our new numerical traceback analysis tool to a core sample of 25 carefully vetted members of $\beta$PMG using Gaia Data Release 3 (DR3) data products and other kinematic surveys. Our method yields a corrected age of 20.4 $\pm$ 2.5 Myr, bridging the gap between kinematic ages (11$-$19 Myr) and other age-dating methods, such as isochrones and lithium depletion boundary (20$-$26 Myr). We explore several association size metrics that can track the spatial extent of $\beta$PMG over time, and we determine that minimizing the variance along the heliocentric curvilinear coordinate $\xi^{\prime}$ (i.e., toward the Galactic Center) offers the least random and systematic errors, due to the wider UVW space velocity dispersion of members of $\beta$PMG along the U-axis, which tends to maximize the spatial growth of the association along the $\xi^{\prime}$-axis over time.