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The VMC survey - XVII. The proper motion of the Small Magellanic Cloud and of the Milky Way globular cluster 47 Tucanae

In this study we use multi-epoch near-infrared observations from the VISTA survey of the Magellanic Cloud system (VMC) to measure the proper motion of different stellar populations in a tile of 1.5 deg sq. in size in the direction of the Galactic globular cluster 47 Tuc. We obtain the proper motion of the cluster itself, of the Small Magellanic Cloud (SMC), and of the field Milky Way stars. Stars of the three main stellar components are selected from their spatial distribution and their distribution in colour-magnitude diagrams. Their average coordinate displacement is computed from the difference between multiple Ks-band observations for stars as faint as Ks=19 mag. Proper motions are derived from the slope of the best-fitting line among 10 VMC epochs over a time baseline of ~1 yr. Background galaxies are used to calibrate the absolute astrometric reference frame. The resulting absolute proper motion of 47 Tuc is (mu_alpha cos(delta), mu_delta)=(+7.26+/-0.03, -1.25+/-0.03) mas/yr. This measurement refers to about 35000 sources distributed between 10 and 60 arcmin from the cluster centre. For the SMC we obtain (mu_alpha cos(delta), mu_delta)=(+1.16+/-0.07, -0.81+/-0.07) mas/yr from about 5250 red clump and red giant branch stars. The absolute proper motion of the Milky Way population in the line-of-sight (l =305.9, b =-44.9) of this VISTA tile is (mu_alpha cos(delta), mu_delta)=(+10.22+/-0.14, -1.27+/-0.12) mas/yr and results from about 4000 sources. Systematic uncertainties associated to the astrometric reference system are 0.18 mas/yr. Thanks to the proper motion we detect 47 Tuc stars beyond its tidal radius.