An extremely low-density exoplanet spins slow

We present constraints on the shape of Kepler-51d, which is a super-puff with a mass $\sim6\,M_\oplus$ and a radius $\sim9\,R_\oplus$, based on detailed modeling of the transit light curve from JWST NIRSpec. The projected shape of this extremely low-density planet is consistent with being spherical, and a projected oblateness $f_\perp>0.2$ can be excluded regardless of the spin obliquity angles. If this is taken as the limit on the true shape of the planet, Kepler-51d is rotating at $\lesssim 50\%$ of its break-up spin rate, or its rotation period is $\gtrsim 33\,$hr. In the more plausible situation that the planetary spin is aligned with its orbital direction to within $30^\circ$, then its oblateness is $<0.08$, which corresponds to a dimensionless spin rate $\lesssim30\%$ of the break-up rotation and a dimensional rotation period $\gtrsim 53\,$hr. This seems to contradict the theoretical expectation that planets with such low masses may be spinning near break-up. We point out the usefulness of the stellar mean density and the orbital eccentricity in constraining the shape of the transiting planet, so planets with well-characterized host and orbital parameters are preferred in the detection of planetary oblateness with the JWST transit method.