The dependence of the magnetic field strength of a sunspot umbra on its intensity and brightness temperature

A partial decrease of the convective energy by the magnetic field of a sunspot causes the temperature of the magnetized plasma to decrease and this leads to a magneto-hydrostatic equilibrium. Thus, we expect a strong relation between magnetic field strength of a sunspot and its temperature/brightness. Here, we investigate this relation in the umbra of the large sunspot in NOAA10930 using spectro-polarimetric data recorded by Solar Optical Telescope on board Hinode. The magnetic field strength is measured by two methods: 1) using the Stokes V profile of the neutral iron 630.25 nm line, and 2) running SIR inversion code on the full Stokes profiles of the two neutral iron 630.15 and 630.25 nm lines. The general behavior is the decreasing of magnetic field strength with temperature/intensity. For intensities less than about 0.2I ph (brightness temperature 4280 K; I ph is the mean photospheric intensity around the sunspot), decreasing magnetic field strength with increasing intensity has a steeper gradient, a power law function clearly explains the non-linear variations of magnetic field strength versus temperature/intensity. For intensities larger than 0.2I ph, at a given intensity, the range of the deviation of magnetic field strength is as large as 0.5 kG. By looking for the location of these points on the studied umbral region, we find that these points are belong to the inner penumbra where penumbral filaments intrude into the dark umbra at two corners of the studied umbral region which shows different magnetic field strengths. This difference can be due to the difference between the evolutionary stages of the penumbral filaments at the two corners during the observation period. For intensities less than 0.2I ph, the observed deviation around the power law behavior is partially due to the presence of umbral dots. Since umbral dots are brighter than the umbra, and the layers of optical depth unity are moved to higher heights, a smaller magnetic field strength may be measured in umbral dots. This causes the data points shift to larger intensities and smaller magnetic field strengths on the corresponding scatter plot. Since the increase in intensity and the decrease in magnetic field strength on different umbral dots are different, the scattering around the power law is explained.