Relations between temperature, T, and optical depth, tau, are often used for describing the photospheric transition from optically thick to optically thin in stellar structure models. We show that this is well justified, but also that currently used
T(tau) relations are often inconsistent with their implementation. As an outer boundary condition on the system of stellar structure equations, T(tau) relations have an undue effect on the overall structure of stars. In this age of precision asteroseismology, we need to re-assess both the method for computing and for implementing T(tau) relations, and the assumptions they rest on. We develop a formulation for proper and consistent evaluation of T(tau) relations from arbitrary 1D or 3D stellar atmospheres, and for their implementation in stellar structure and evolution models. We extract radiative T(tau) relations, as described by our new formulation, from 3D simulations of convection in deep stellar atmospheres of late-type stars from dwarfs to giants. These simulations employ realistic opacities and equation of state, and account for line-blanketing. For comparison, we also extract T(tau) relations from 1D MARCS model atmospheres using the same formulation. T(tau)-relations from our grid of 3D convection simulations display a larger range of behaviours with surface gravity, compared with those of conventional theoretical 1D hydrostatic atmosphere models. Based on this, we recommend no longer to use scaled solar T(tau) relations. Files with T(tau) relations for our grid of simulations are made available to the community, together with routines for interpolating in this irregular grid. We also provide matching tables of atmospheric opacity, for consistent implementation in stellar structure models.
We study the effects of different descriptions of the solar surface convection on the eigenfrequencies of p-modes. 1-D evolution calculations of the whole Sun and 3-D hydrodynamic and magnetohydrodynamic simulations of the current surface are perform
ed. These calculations rely on realistic physics. Averaged stratifications of the 3-D simulations are introduced in the 1-D solar evolution or in the structure models. The eigenfrequencies obtained are compared to those of 1-D models relying on the usual phenomenologies of convection and to observations of the MDI instrument aboard SoHO. We also investigate how the magnetic activity could change the eigenfrequencies and the solar radius, assuming that, 3 Mm below the surface, the upgoing plasma advects a 1.2 kG horizontal field. All models and observed eigenfrequencies are fairly close below 3 mHz. Above 3 mHz the eigenfrequencies of the phenomenological convection models are above the observed eigenfrequencies. The frequencies of the models based on the 3-D simulations are slightly below the observed frequencies. Their maximum deviation is ~ 3 mu Hz at 3 mHz but drops below 1 mu Hz at 4 mHz. Replacing the hydrodynamic by the magnetohydrodynamic simulation increases the eigenfrequencies. The shift is negligible below 2.2 mHz and then increases linearly with frequency to reach ~ 1.7 mu Hz at 4 mHz. The impact of the simulated activity is a 14 milliarcsecond shrinking of the solar layers near the optical depth unity.
