Slow Rotation for the Super-Puff Planet Kepler-51d

Super-puffs are low-density planets of unknown origin and composition. If they form by accreting nebular gas through a circumplanetary disk, one might expect super-puffs to be spinning quickly. Here, we derive upper limits on the rotational oblateness of the super-puff Kepler-51d, based on precise transit observations with the NIRSpec instrument aboard the James Webb Space Telescope. The absence of detectable oblateness-related anomalies in the light curve leads to an upper limit of about $0.15$ on the planet's sky-projected oblateness. Assuming the sky-projected oblateness to be representative of the true oblateness, the rotation period of Kepler-51d is $\gtrsim 40$ hours, or equivalently, its rotation speed is $\lesssim 42\%$ of the breakup speed. Alternatively, if the apparently low density of Kepler-51d is due to an opaque planetary ring, the ring must be oriented within $30\deg$ of face-on and have an inner radius smaller than $1.2$ times the planet's radius. Separately, the lack of anomalies exceeding $0.01\%$ in the ingress and egress portions of the light curve places a constraint on the model of Wang & Dai, in which the planet's apparently low density is due to a dusty outflowing atmosphere.