The solar system gas giant planets are oblate due to their rapid rotation. A measurement of the planet's projected oblateness would constrain the planet's rotational period. Planets that are synchronously rotating with their orbital revolution will be rotating too slowly to be significantly oblate; these include planets with orbital semimajor axes 0.2 AU (for MP ~ MJupiter and M* ~ M☉). Jupiter-like planets in the range of orbital semimajor axis 0.1 to 0.2 AU will tidally evolve to synchronous rotation on a timescale similar to main-sequence stars' lifetimes. In this case, an oblateness detection will help constrain the planet's tidal Q value. The projected oblateness of a transiting extrasolar giant planet is measurable from a very high photometric precision transit light curve. For a Sun-sized star and a Jupiter-sized planet, the normalized flux difference in the transit ingress/egress light curve between a spherical and an oblate planet is a few to 15 × 10-5 for oblateness similar to Jupiter and Saturn, respectively. The transit ingress and egress are asymmetric for an oblate planet with orbital inclination different from 90° and a nonzero projected obliquity. A photometric precision of 10-4 has been reached by Hubble Space Telescope (HST) observations of the known transiting extrasolar planet HD 209458b. Kepler, a NASA discovery-class mission designed to detect transiting Earth-sized planets, requires a photometric precision of 10-5 and expects to find 30 to 40 transiting giant planets with orbital semimajor axes 0.2 AU. Furthermore, part-per-million photometric precision (reached after averaging over several orbital periods) is expected from three other space telescopes to be launched within the next three years. Thus, an oblateness measurement of a transiting giant planet is realistic in the near future.

Constraining the Rotation Rate of Transiting Extrasolar Planets by Oblateness Measurements

(Abridged) The solar system gas giant planets are oblate due to their rapid rotation. A measurement of the planet's projected oblateness would constrain the planet's rotational period. Planets that are synchronously rotating with their orbital revolution will be rotating too slowly to be significantly oblate; these include planets with orbital semi-major axes <~ 0.2AU (for M_P ~ M_J and M_* ~ M_sun). Jupiter-like planets in the range of orbital semi-major axes 0.1 AU to 0.2 AU will tidally evolve to synchronous rotation on a timescale similar to main sequence stars' lifetimes. In this case an oblateness detection will help constrain the planet's tidal Q value. 
The projected oblateness of a transiting extrasolar giant planet is measurable from a very high-photometric-precision transit light curve. For a sun-sized star and a Jupiter-sized planet the normalized flux difference in the transit ingress/egress light curve between a spherical and an oblate planet is a few to 15 x 10^{-5} for oblateness similar to Jupiter and Saturn respectively. The transit ingress and egress are asymmetric for an oblate planet with an orbital inclination different from 90 degrees and a non-zero projected obliquity. A photometric precision of 10^{-4} has been reached by HST observations of the known transiting extrasolar planet HD209458b. In addition several planned space missions (including MOST, MONS, Corot, and Kepler) will study or find transiting giant planets with photometric precision 0.2 AU.

https://arxiv.org/pdf/astro-ph/0204225.pdf

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Constraining the Rotation Rate of Transiting Extrasolar Planets by Oblateness Measurements

Constraining the Rotation Rate of Transiting Extrasolar Planets by Oblateness Measurements

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