We show that the gas giant exoplanet HD 189733b is less oblate than Saturn , based on Spitzer Space Telescope photometry of seven transits .
The observable manifestations of oblateness would have been slight anomalies during the ingress and egress phases , as well as variations in the transit depth due to spin precession .
Our nondetection of these effects gives the first empirical constraints on the shape of an exoplanet .
The results are consistent with the theoretical expectation that the planetary rotation period and orbital period are synchronized , in which case the oblateness would be an order of magnitude smaller than our upper limits .
Conversely , if HD 189733b is assumed to be in a synchronous , zero-obliquity state , then the data give an upper bound on the quadrupole moment of the planet ( J _ { 2 } < 0.068 with 95 \% confidence ) that is too weak to constrain the interior structure of the planet .
An Appendix describes a fast algorithm for computing the transit light curve of an oblate planet , which was necessary for our analysis .