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We investigate the prospects for characterizing extrasolar giant planets bymeasuring planetary oblateness from transit photometry and inferring planetaryrotational periods. The rotation rates of planets in the solar system varywidely, reflecting the planets' diverse formational and evolutionary histories.A measured oblateness, assumed composition, and equation of state yields arotation rate from the Darwin-Radau relation. The lightcurve of a transitingoblate planet should differ significantly from that of a spherical one with thesame cross-sectional area under identical stellar and orbital conditions.However, if the stellar and orbital parameters are not known a priori, fittingfor them allows changes in the stellar radius, planetary radius, impactparameter, and stellar limb darkening parameters to mimic the transit signatureof an oblate planet, diminishing the oblateness signature. Thus even ifHD209458b had an oblateness of 0.1 instead of our predicted 0.003, it wouldintroduce a detectable departure from a model spherical lightcurve at the levelof only one part in 10^5. Planets with nonzero obliquity break this degeneracybecause their ingress lightcurve is asymmetric relative to that from egress,and their best-case detectability is of order 10^{-4}. However, the measuredrotation rate for these objects is non-unique due to degeneracy betweenobliquity and oblateness and the unknown component of obliquity along the lineof sight. Detectability of oblateness is maximized for planets transiting nearan impact parameter of 0.7 regardless of obliquity. Future measurements ofoblateness will be challenging because the signal is near the photometriclimits of current hardware and inherent stellar noise levels.Comment: 13 pages -- accepted to the Astrophysical Journa

Measuring the Oblateness and Rotation of Transiting Extrasolar Giant Planets