The projection of the angle between the stellar spin and the planetary orbit can be measured for transiting planets, by following the radial-velocity anomaly during transit.
The figure below traces the changing mood of the community on the topic. The consensus view before 2008 was that hot Jupiter orbits should be mostly aligned with the rotation of the star because they spiraleld closer to the star in a gaseous disc. As of early 2012, it is believed that the orbits of close-in planets are randomly oriented because they were flung inwards by violent interaction with another planet or a companion star.
Up to 2008, the spin-orbit angle was measured for a handful of systems, including HD 209458, HD 189733 and TrES-1, and all were compatible with aligned orbits within a few degrees.
Then Narita et al. (2008) announced a tentative detection of misalignment for HD 17156, a result that later turned out to be incorrect and due to instrumental systematics (Cochran et al. 2008, Narita et al. 2009). A few months later though, Hébrard et al. (2008) found a possible misalignment for the XO-3 system, and this was later confirmed with more accurate measurements by Winn et al. (2009).
Why all these results “possible” and “tentative”? The Rossiter-McLaughlin measurements required to determine the spin-orbit angle use the spectrographs and procedures designed for radial-velocity planet-search programmes. However, while these measurements have very high long-term stability, they are not optimized for the RM measurements, which target fainter objects and concentrate all measurements on the short timescale of the transit event. Instrumental effects in this regime – faint objects and hour timescales – can be strong and are poorly characterized because they do not affect the standard planet searches.
Following the XO-3 case, in a bold application of probability theory, Fabrycky & Winn (2009) inferred a bimodal distribution of spin-orbit angle from this small sample and a single misaligned system.
The case of XO-3 could have been an anomaly, because XO-3b is a very heavy planet (12 Jupiter masses). However, detections of misaligned orbit followed for several other objects. While the extremely eccentric HD 80606b (Pont et al. 2009) could be construed as another exception, nothing seemed special about the misaligned WASP-14 (Johnson et al. 2009) , HAT-P-7 (Narita et al. 2009, Winn et al. 2009) or CoRoT-1 (Pont et al. 2010).
The new measurements revealed not only misaligned systems, but entirely retrograde orbits. Aligned systems, though, were turning up as well, like HAT-P-2 (Loeillet et al. 2008) or CoRoT-2 (Bouchy et al. 2008).
Winn et al. (2010) put rhyme and reason in this seemingly chaotic situation, by showing that all the aligned and misaligned systems were neatly separated according to the temperature of their host star, with cooler stars having aligned spins and orbits. The idea is that stars cool enough to have a convective envelope would have generated enough tidal interactions to align the stellar spin and planetary orbital plane, while hotter stars with radiative envelopes would have preserved the initial spin-orbit angle (the presence of a convective envelope is expected to sharply increases the amplitude of tidal effects, because it slows down the adjustment of the star to the rotating gravity field of the planet and therefore increases the lag angle of the tidal bulges).
If this is correct, then the stunning implication is that when the effect of tidal alignment is removed, the spin and orbits are generally misaligned, with the angle between the two almost random! In other words, either the orbits of hot Jupiters have lost all memory of the position of the disc during their inwards migration, or the star itself has lost its alignment with the protoplanetary disc. The first case seems more likely, and provides strong support for the hypothesis that hot Jupiters were brought close to their star by interaction with another planet, rather than migrating inwards by tidal interaction with the gas disc.
The new measurements of spin-orbit angles since 2010 have largely supported the radiative/convective dichotomy, with no clear exception yet out of more than two dozen cases.