Issue 
A&A
Volume 555, July 2013



Article Number  A75  
Number of page(s)  9  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201321545  
Published online  04 July 2013 
Online material
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 zaxis, 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 boundfree transitions, freefree transitions, Thomson scattering, and Rayleigh scattering. The most important source for the solar photosphere comes from boundfree 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 boundfree transitions , where σ is a constant that depends on the atom/ion involved in the boundfree 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 ℐ_{0}, 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) = D_{0} + D′(r,t) and substitute in Eq. (A.2) to find (A.3)where we neglected higherorder terms. Because the optical depth τ_{ν} is constant, the zerothorder terms in Eq. (A.3) show (A.4)To find D′ we look at the firstorder 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 D_{0} 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
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