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Appendix A: Surface with constant optical depth

The optical depth τ_ν can be calculated using (Rybicki & Lightman 1986; Rutten 2003) (A.1)This shows that the optical depth depends on the frequency ν, the LOS, and the extinction coefficient α, which also depends on the photon frequency. To simplify the calculations we assume that the LOS is directed along the z-axis, i.e. ds = dz. The goal is to find the height D for which the optical depth at a given frequency τ_ν is constant. Because we are using observations of the photosphere, we assume that the intensity observations are taken in the continuum and not in the line core. In Rutten (2003) we can find different sources for opacity in the continuum each with their own extinction coefficient. We have bound-free transitions, free-free transitions, Thomson scattering, and Rayleigh scattering. The most important source for the solar photosphere comes from bound-free transitions where the hydride H⁻ absorb radiation to form hydrogen and a free electron (Marshall 2003; Pradhan & Nahar 2011). From Rutten (2003) we find that for bound-free transitions , where σ is a constant that depends on the atom/ion involved in the bound-free transition, n_i is the number density in the ionising level, h is the Planck constant, k_B is the Boltzmann constant, T is the temperature of the plasma, and ν is the frequency. We assume that n_i is proportional to the density ρ, thus the monochromatic optical depth is where a indicates the fraction of ionisation. We can now linearise both the density and the temperature to find where ℐ₀, A(r), and B(r) are given by

The optical depth (Eq. (A.1)) is thus given by (A.2)In general, D is a function of r,ϕ, and t, but since we are dealing with axisymmetric modes, we have D(r,t). We linearise D(r,t) = D₀ + D′(r,t) and substitute in Eq. (A.2) to find (A.3)where we neglected higher-order terms. Because the optical depth τ_ν is constant, the zeroth-order terms in Eq. (A.3) show (A.4)To find D′ we look at the first-order terms in Eq. (A.3) and find Since both P′ and ξ_r are proportional to exp{i(kz − ωt)}, (A.5)The equation for a surface with constant optical depth is given by g(r,z,t) ≡ z − D(r,t). In the uniform case we have D₀ constant, A(r) constant, and B(r) = 0, thus g(r,z,t) has a similar form as the surface that follows the motions of the plasma (i.e. f(r,ϕ,z,t) ≡ z − ξ_z(r,ϕ,z = 0,t)). We calculate the normal to the surface gWe relate a surface element dS to an elemental surface element in the horizontal plane dS_h, i.e. which in a linear approximation is dS = dS_h = rdrdϕ. This shows that integrating over the surface where τ_ν is constant or integrating over the surface that follows the motions of the plasma is identical, since for a uniform equilibrium and in a linear expansion the elemental surface elements are the same.

© ESO, 2013