We present precise radial-velocity measurements of WASP-1 and WASP-2 throughout transits of their giant planets. Our goal was to detect the Rossiter-McLaughlin (RM) effect, the anomalous radial velocity observed during eclipses of rotating stars, which can be used to study the obliquities of planet-hosting stars. For WASP-1 a weak signal of a prograde orbit was detected with ~2sigma confidence, and for WASP-2 no signal was detected. The resulting upper bounds on the RM amplitude have different implications for these two systems, because of the contrasting transit geometries and the stellar types. Because WASP-1 is an F7V star, and such stars are typically rapid rotators, the most probable reason for the suppression of the RM effect is that the star is viewed nearly pole-on. This implies the WASP-1 star has a high obliquity with respect to the edge-on planetary orbit. Because WASP-2 is a K1V star, and is expected to be a slow rotator, no firm conclusion can be drawn about the stellar obliquity. Our data and our analysis contradict an earlier claim that WASP-2b has a retrograde orbit, thereby revoking this systems status as an exception to the pattern that cool stars have low obliquities.
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