Opacity Rosseland mean

Since a vast number of data are required to calculate the contribution to the Rosseland mean from many radiative processes, the calculation of stellar opacities is a long and involved process. Also, since the total opacity (Rosseland mean) is an harmonic mean, the opacity must be recalculated for every chemical mixture thus... 

The effective opacity is given by taking the wavelength-dependent opacity of the material at the relevant temperature and density and forming the (harmonic) Rosseland mean ... 

Derive the formula (5.25) for the Rosseland mean opacity by comparing the transfer equation for K, Eq. (3.15), for monochromatic and integrated radiation. 

Show that, if the monochromatic opacity varies as v 3T 1/2 over relevant frequency ranges, the Rosseland mean varies as T 3-5, as in Eq. (5.26). 

F. Palla. Low-temperature Rosseland mean opacities for zero-metal gas mixtures. In G. H. F. Diercksen, W. F. Huebner, and P. W. Langhoff, eds., Molecular Astrophysics - State of the Art and Future Directions, p. 687, D. Reidel, Dordrecht, 1985. 

For theories of planet formation and disk chemistry, the mid-plane temperature, Tm, is more important than the surface temperature. Also, to solve Eq. (3.4) or Eq. (3.6) we need to know v, which depends on Tm, not on Te. For irradiation-dominated disks Tm is roughly equal to the surface effective temperature Te. For disks in which accretional heating is important this is not the case. A rough estimate for the mid-plane temperature valid for any case is T4 = (3/8)T4viscrR + r4irr, where T,visc and 7). T are given above and tr = E/cr with /cr the Rosseland mean opacity of the gas-dust mixture at the temperature Tm. The expression for Tm is therefore not explicit, as it is also used for kr. For realistic dust opacities it requires a numerical iterative solution procedure. For a gray opacity, or for piecewise power-law opacities, such as those in Bell et al. (1997), it can be solved analytically. 

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