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Limb darkening measurements from TESS and Kepler light curves of transiting exoplanets

Inaccurate limb-darkening models can be a significant source of error in the analysis of the light curves for transiting exoplanet and eclipsing binary star systems. To test the accuracy of published limb-darkening models, I have compared limb-darkening profiles predicted by stellar atmosphere models to the limb-darkening profiles measured from high-quality light curves of 43 FGK-type stars in transiting exoplanet systems observed by the Kepler and TESS missions. The comparison is done using the parameters $h^{\prime}_1 = I_λ(\frac{2}{3})$ and $h^{\prime}_2 = h^{\prime}_1 - I_λ(\frac{1}{3})$, where $I_λ(μ)$ is the specific intensity emitted in the direction $μ$, the cosine of the angle between the line of sight and the surface normal vector. These parameters are straightforward to interpret and insensitive to the details of how they are computed. I find that most (but not all) tabulations of limb-darkening data agree well with the observed values of $h^{\prime}_1$ and $h^{\prime}_2$. There is a small but significant offset $Δh^{\prime}_1 \approx 0.006$ compared to the observed values that can be ascribed to the effect of a mean vertical magnetic field strength $\approx 100$\,G that is expected in the photospheres of these inactive solar-type stars but that is not accounted for by typical stellar model atmospheres. The implications of these results for the precision of planetary radii measured by the PLATO mission are discussed briefly.