Determination of the Alfvén Speed and Plasma-beta Using the Seismology of Sunspot Umbra

For 478 centrally located sunspots observed in the optical continuum with Solar Dynamics Observatory/Helioseismic Magnetic Imager, we perform seismological diagnostics of the physical parameters of umbral photospheres. The new technique is based on the theory of slow magnetoacoustic waves in a non-isothermally stratified photosphere with a uniform vertical magnetic field. We construct a map of the weighted frequency of three-minute oscillations inside the umbra and use it for the estimation of the Alfvn speed, plasma-beta, and mass density within the umbra. We find the umbral mean Alfvn speed ranges between 10.5 and 7.5 km s(-1) and is negatively correlated with magnetic field strength. The umbral mean plasma-beta is found to range approximately between 0.65 and 1.15 and does not vary significantly from pores to mature sunspots. The mean density ranges between (1-6). x 10(-4) kg m(-3) and shows a strong positive correlation with magnetic field strength.