We report results from 2 1/2 yr of high precision radial velocity (RV) monitoring of Barnards star. The high RV measurement precision of the VLT-UT2+UVES of 2.65 m/s made the following findings possible. (1) The first detection of the change in the RV of a star caused by its space motion (RV secular acceleration). (2) An anti-correlation of the measured RV with the strength of the filling-in of the Halpha line by emission. (3) Very stringent mass upper limits to planetary companions. Using only data from the first 2 years, we obtain a best-fit value for the RV secular acceleration of 5.15+/-0.89 m/s/yr. This agrees with the predicted value of 4.50 m/s/yr based on the Hipparcos proper motion and parallax combined with the known absolute radial velocity of the star. When the RV data of the last half-year are added the best-fit slope is strongly reduced to 2.97+/-0.51 m/s/yr suggesting the presence of additional RV variability in the star. Part of it can be attributed to stellar activity as we demonstrate by correlating the residual RVs with an index that describes the filling-in of the Halpha line by emission. A correlation coefficient of -0.50 indicates that the appearance of active regions causes a blueshift of photospheric absorption lines. Assuming that active regions basically inhibit convection we discuss the possibility that the fundamental (inactive) convection pattern in this M4V star produces a convective redshift. We also determine upper limits to the projected mass msini and to the true mass m of hypothetical planetary companions in circular orbits. For the separation range 0.017-0.98 AU we exclude any planet with msini>0.12 Mjupiter and m>0.86 Mjupiter. Throughout the habitable zone, i.e. 0.034-0.082 AU, we exclude planets with msini>7.5 Mearth and m>3.1 Mneptune.
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