To investigate the change of WASP-121 b’s transmission spectrum induced by its rotation, we integrated the NIRSpec light curves over bins of 244 pixel columns of the NRS1 and 237 pixel columns of the NRS2 detector to generate 14 spectroscopic light curves. As in the white light curve fit, we fit these light curves simultaneously, adopting the same eccentricity, argument of periapsis and period as before and fitting for b and a/R∗which are shared between all light curves. All other parameters are different for all light curves. Again, we first ran a least-squares fit and sampled the parameters’ posterior distributions using an MCMC. In the MCMC, we initiated 500 walkers in a tight Gaussian ball around the least-squares solution and ran 100,000 steps for each walker, discarding the first 50,000 steps as burn-in. The final samples were thinned by a factor of 200.

We applied three modeling approaches to fit the spectroscopic light curves in order to estimate the model assumptions’ impacts on the inferred planet radius coefficients Ri/R∗. These three approaches were 1. R2/R∗= 0 and wide uniform priors on the limb darkening coefficients, 2. R2/R∗as a free parameter and wide uniform priors on the limb darkening coefficients, and 3. R2/R∗as a free parameter and Gaussian priors on the limb darkening coefficients which we adopted from the exotic-LD package [51] using the ‘stagger’ grid of stellar models [52].
The MCMCs’ posteriors of the planet radius polynomial coefficients from these three modeling approaches analyzing the data from the Firefly reduction and modeling approach number one applied to the Eureka! reduction are plotted in Extended Data Fig. 5. They reveal that the R1/R∗coefficient, which gives the linear rate of the planet’s radius change relative to the star’s radius during transit, is indifferent toward the choice of priors for the limb darkening. This is because R1/R∗is the coefficient for a polynomial term that is odd about the point of conjunction. As such, a nonzero R1/R∗will induce a transit shape that is asymmetric about the point of conjunction which cannot be replicated by changing the limb darkening or the orbital parameters. When we adopt R2/R∗= 0 in Mrot, the measurements of R1/R∗again do not change (see Extended Data Fig. 5) and thus, our measurements of R1/R∗are robust against choices of limb darkening priors and the value of R2/R∗. In contrast to R1/R∗, the measurements reveal that R0/R∗, giving the planet-tostar radius ratio at the point of conjunction, and R2/R∗, describing the curvature of the radius function in time, are both sensitive to modeling approaches of the stellar limb darkening, as their uncertainties greatly increase when moving from tight Gaussian priors to uninformative uniform priors on u1 and u2. The reason for this degeneracy is that R2/R∗describes an even term about the point of conjunction and is correlated with R0/R∗(see Extended Data Fig. 2). Any even changes about the point of conjunction (such as a decrease of the radius in the first and a converse increase in the second half of the transit) has a similar effect on the light curve as the limb darkening that is symmetric about the center of the stellar disk. Therefore, R2/R∗and the correlated R0/R∗cannot be reliably disentangled from the limb darkening using the JWST data.