Source Depth of Three-minute Umbral Oscillations

We infer the depth of the internal sources giving rise to three-minute umbral oscillations. Recent observations of ripple-like velocity patterns of umbral oscillations supported the notion that there exist internal sources exciting the umbral oscillations. We adopt the hypothesis that the fast magnetohydrodynamic (MHD) waves generated at a source below the photospheric layer propagate along different paths, reach the surface at different times, and excited slow MHD waves by mode conversion. These slow MHD waves are observed as the ripples that apparently propagate horizontally. The propagation distance of the ripple given as a function of time is strongly related to the depth of the source. Using the spectral data of the Fe i 5435 Å line taken by the Fast Imaging Solar Spectrograph of the Goode Solar Telescope at Big Bear Solar Observatory, we identified five ripples and determined the propagation distance as a function of time in each ripple. From the model fitting to these data, we obtained the depth between 1000 and 2000 km. Our result will serve as an observational constraint to understanding the detailed processes of magnetoconvection and wave generation in sunspots.