6.3 p-mode frequencies

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Figure 42: Pressure as a function of depth for an averaged 3D model (full drawn), for a standard 1D solar model (dashed) and for a 3D model with turbulent pressure removed (dash-dot). Turbulent pressure and the hiding of hot gas contribute about equally to raising the photosphere about one scale height.

There is a discrepancy between the observed p-mode frequencies and the eigenfrequencies calculated from one-dimensional models. Part of this difference occurs because convection enlarges the p-mode resonant cavity and lowers the frequencies of the higher frequency modes with turning points in the photosphere (Figure 42View Image). This is due to two effects: First, the turbulent pressure is about 15% of the gas pressure near the top of the convection zone and the extra pressure support raises the atmosphere about half a scale height. Second, because of the high temperature sensitivity of the H opacity, one does not see the hotter gas at the surface. Optical depth unity lies higher in hotter regions where the temperature decreasing outward is lower. Thus the horizontally averaged temperature is actually higher than obtained with a 1D model having the same effective temperature. The higher temperature means the scale height is larger, which raises the atmosphere another half scale height. The total effect is an atmosphere that is extended by a scale height compared to 1D models (Figure 42View Image). The larger cavity reduces the discrepancy between the observed and theoretical mode frequencies as calculated from 1D models (Figure 43View Image) (Rosenthal et al., 1999). The frequency residuals of the f-mode (which is nearly independent of the hydrostatic structure) are unchanged. The residuals of the p-modes from the simulation now are the same order of magnitude as for the f-mode and change sign. This indicates that the remaining discrepancies are due to mode physics effects and depend on depth or frequency or both.

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Figure 43: Frequency residuals (observations-calculated) scaled by the ratio Qnℓ of the mode mass to the mode mass of the radial mode with the same frequency. On the left frequencies calculated from the standard 1D solar model S of Christensen-Dalsgaard et al. (1996) and on the right calculated from the horizontal and time averaged 3D simulation extended with a matched mixing length model into the deeper solar layers.

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