Résumé
Context. The radius of a star is a very important constraint to evolutionary models, particularly when combined with asteroseismology. Diameters can now be measured interferometrically with great precision (better than 1%), but the center-to-limb darkening (CLD) remains a potential source of bias. Measuring this bias is possible by completely resolving the star using long-baseline interferometry, and has only been achieved for a handful of stars. Aims: The red giant eta Ser (K0III-IV) is a particularly interesting target, as asteroseismic oscillations have recently been detected in this star by spectroscopy. We aim to measure its radius with high accuracy, debiased from limb darkening, in order to bring new constraints to its models. Methods: We obtained interferometric observations of eta Ser in the near-infrared using the CHARA/FLUOR instrument, in particular in the so-called second lobe of visibility in order to constrain the CLD and debias our diameter estimation. Results: The limb darkened angular diameter of eta Ser is 2.944 ± 0.010 mas (using spherical photosphere models PHOENIX and MARCS for the limb darkening), that converts into a radius of 5.897 ± 0.028 R_ with the Hipparcos parallax. Thanks to a precise visibility measurement in the second lobe of the visibility function of eta Ser and a one-parameter limb-darkened visibility profile, we were able to show that the photosphere models have the best agreement possible. Conclusions: Our limb darkening measurement of eta Ser is in agreement with existing atmosphere models of this star, with a slightly better agreement for models using spherical geometry. This is a strong indication that interferometric angular diameter measurements for red giants, corrected for the CLD using models, are unbiased at a very small level (a fraction of 1%). In particular, this strengthens our confidence in the existing catalogues of calibrator stars for interferometry that are based on giant stars similar to eta Ser. The high accuracy of our measurement brings a new and strong constraint for the asteroseismic modeling of this star.