Tests of stellar model atmospheres by optical interferometry. VLTI/VINCI limb-darkening measurements of the M4 giant ψ Phe

We present K-band interferometric measurements of the limb-darkened (LD) intensity profile of the M 4 giant star ψ Phoenicis obtained with the Very Large Telescope Interferometer (VLTI) and its commissioning instrument VINCI. High- precision squared visibility amplitudes in the second lobe of the visibility function were obtained employing two 8.2 m Unit Telescopes (UTs). This took place one month after light from UTs was first combined for interferometric fringes. In addition, we sampled the visibility function at small spatial frequencies using the 40 cm test siderostats. Our measurement constrains the diameter of the star as well as its center-to-limb intensity variation (CLV). We construct a spherical hydrostatic PHOENIX model atmosphere based on spectrophotometric data from the literature and compare its CLV prediction with our interferometric measurement. We compare as well CLV predictions by plane-parallel hydrostatic PHOENIX, ATLAS9 ,a ndATLAS12 models. We find that the Rosseland angular diameter as predicted by comparison of the spherical PHOENIX model with spectrophotometry is in good agreement with our interferometric diameter measurement. The shape of our measured visibility function in the second lobe is consistent with all considered PHOENIX and ATLAS model predictions, and is significantly different to uniform disk (UD) and fully darkened disk (FDD) models. We derive high-precision fundamental parameters for ψ Phe, namely a Rosseland angular diameter of 8.13 ± 0.2 mas, with the Hipparcos parallax corresponding to a Rosseland linear radius R of 86 ± 3 R� , and an effective temperature of 3550 ± 50 K, with R corresponding to a luminosity of log L/L� = 3.02 ± 0.06. Together with evolutionary models, these values are consistent with a mass of 1.3 ± 0.2 M� , and a surface gravity of log g = 0.68 ± 0.11.