black-gray line in Figure \[fig:spatial\]] is coincident with the semi-minor axis of the best fit-ellipses, i.e. perpendicular to the semi-major axis plotted in Figure \[fig:spatial\].

Rotation in the plane of the sky {#subsec:rot}
--------------------------------

### Rotation profile {#subsec:vprofile}

In this subsection, we analyze the average internal motions of Fe-poor and Fe-rich stars as a function of the radial distance from the cluster center. As a first step, we converted the $\mu_{\rm \alpha}\cos \delta$ and $\mu_{\rm \delta}$ components of proper motions into a radial ($\mu_{\rm RAD}$) and a tangential ($\mu_{\rm TAN}$) motions on the plane of the sky, correcting for the effect of perspective expansion/contraction as in @vandenven2006. We divided the cluster fields of $\omega\,$Centauri and M22 into 17 and 6 circular annulii, respectively, determined with the naive estimator method [@silverman1986]. To account for the different star densities at different distances from the cluster center we used wider bins in the outskirt of the cluster (in the case of $\omega\,$Centauri: 5 arcmin for the innermost bin, 3.3 arcmin for $1.2R_{\rm h}< r < 3.2R_{\rm h}$, and 4.2 arcmin for $3.2R_{\rm h}< r < 4.2R_{\rm h}$ and 5.9 arcmin for $r > 4.2R_{\rm h}$).

For each annulus, we used the routines provided 