Center-to-limb observations of the Sun with ALMA

We measured the center-to-limb variation of the brightness temperature, $T_b$, from ALMA full-disk images at two frequencies and inverted the solution of the transfer equation to obtain the electron temperature, $T_e$ as a function of optical depth, $\tau$. The ALMA images are very similar to AIA images at 1600\AA. The brightness temperature at the center of the disk is 6180 and 7250 K at 239 and 100 GHz respectively, with dispersions of 100 and 170 K. Plage regions stand out clearly in the 239/100 GHz intensity ratio, while faculae and filament lanes do not. The solar disk radius, reduced to 1 AU, is $961.1\pm2.5$ arcsec and $964.1\pm4.5$ arcsec at 239 and 100 GHz respectively. A slight but statistically significant limb brightening is observed at both frequencies. The inversion of the center-to-limb curves shows that $T_e$ varies linearly with the logarithm of optical depth for $0.34<\tau_{100\,GHz}<12$, with a slope $d\ln T_e/d\tau=-608$ K. Our results are 5% lower than predicted by the average quiet sun model C of Fontenla et al. (1993), but do not confirm previous reports that the mm-$\lambda$ solar spectrum is better fitted with models of the cell interior.