  We investigate the prospects for characterizing extrasolar giant planets by
measuring planetary oblateness from transit photometry and inferring planetary
rotational periods. The rotation rates of planets in the solar system vary
widely, reflecting the planets' diverse formational and evolutionary histories.
A measured oblateness, assumed composition, and equation of state yields a
rotation rate from the Darwin-Radau relation. The lightcurve of a transiting
oblate planet should differ significantly from that of a spherical one with the
same cross-sectional area under identical stellar and orbital conditions.
However, if the stellar and orbital parameters are not known a priori, fitting
for them allows changes in the stellar radius, planetary radius, impact
parameter, and stellar limb darkening parameters to mimic the transit signature
of an oblate planet, diminishing the oblateness signature. Thus even if
HD209458b had an oblateness of 0.1 instead of our predicted 0.003, it would
introduce a detectable departure from a model spherical lightcurve at the level
of only one part in 10^5. Planets with nonzero obliquity break this degeneracy
because their ingress lightcurve is asymmetric relative to that from egress,
and their best-case detectability is of order 10^{-4}. However, the measured
rotation rate for these objects is non-unique due to degeneracy between
obliquity and oblateness and the unknown component of obliquity along the line
of sight. Detectability of oblateness is maximized for planets transiting near
an impact parameter of 0.7 regardless of obliquity. Future measurements of
oblateness will be challenging because the signal is near the photometric
limits of current hardware and inherent stellar noise levels.
