by @vasiliev2019 to compute the median radial ($\Delta \mu_{\rm RAD}$) and tangential ($\Delta \mu_{\rm TAN}$) motions, thus accounting for systematic errors in Gaia DR2 proper motions. Furthermore, *Gaia* DR2 uncertainties on proper motion are underestimated by a factor of $\sim 10\%$ and $\sim 30\%$ for stars with magnitude $G>16$ and $G<13$, respectively.

We therefore artificially increased the observed uncertainties on the proper motions as in @bianchini2019.\
We then estimated the uncertainty relative to each point by bootstrapping with replacements performed 1,000 times.

The radial and tangential proper motions have been converted from mas/yr to km/s, ($\Delta V_{\rm RAD}$ and $\Delta V_{\rm TAN}$) adopting the distances listed in Table \[tab:parameters\], derived in @baumgardt2018. The radial and tangential velocity vs.the distance from the cluster center are shown in Figure \[fig:profiles\]. The radial coordinate has been normalized to the half-light radius, from [@baumgardt2018] (see Table \[tab:parameters\]). A visual inspection of Figure \[fig:profiles\] reveals that the radial profiles of $\omega\,$Centauri and M22 are consistent with a zero-velocity up to at least $\sim$4 and $\sim$2 $R_{\rm h}$ respectively. A possible exception is provided by the Fe-rich population of $\omega\,$Centauri, which seems to exhibit larger radial velocities for distances greater than 4 $R_{\rm h}$.

Concerning the 