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As far as I can tell, we do not yet have the precision to even put reasonable bounds on an exoplanets obliquity, but wikipedia seems to indicate that this may be possible in the "near future." It seems like this would have to be accomplished by direct imaging, either by directly observing rotational flattening of an exoplanets, or by looking for moons and assuming that the planet is tidally locked to the same plane as its satellite.

How close would you estimate we are to this kind of precision? Are there other approaches to measuring planetary obliquity?

Obviously, I'm not expecting a definite answer. Just wondering if anyone knows of any research in this area or has any thoughts about it.

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It may, in the "near" future, be possible to detect color changes in reflected light off of a planet's surface. If cyclical, this could indicate seasonal vegetation changes which could help us determine obliquity. Of course, we may be a little more excited about the vegetation!! – Jack R. Woods Sep 14 '15 at 18:30

I searched and found this paper

that suggest(s) that the Fourier spectrum of a thermal lightcurve may be sufficient to determine the orbital inclination of non-transiting short-period planets, the rotational inclination of stars and brown dwarfs, and the obliquity of directly imaged planets

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