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week3 - 1 Monday October 3 Stellar Atmospheres There exist...

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1 Monday, October 3: Stellar Atmospheres There exist entire books written about stellar atmospheres; I will only give a brief sketch of the simplest approximations used in studying stellar at- mospheres. In particular, I want to discuss that most useful quantity, the Rosseland mean opacity . 1 A photon of frequency ν travels an infinitesimal distance ds inside a star. In that distance, it can be scattered or absorbed. For simplicity, I will assume that the scattering is coherent, and that the energy of an absorbed photon re- emerges as thermal radiation. The radiative transfer equation for scattering is dI ν ds (scattering) = - σ ν I ν + σ ν J ν , (1) where σ ν is the scattering coefficient and J ν is the angle-averaged specific intensity. 2 The radiative transfer equation for absorption is dI ν ds (absorption) = - α ν I ν + α ν S ν , (2) where α ν is the absorption coefficient, and the source function S ν is equal to the Planck function B ν ( T ) if, as we’ve assumed, the energy of the absorbed photons is converted to thermal radiation. Combining the effects of absorption and scattering, we find dI ν ds = - ( α ν + σ ν ) I ν + ( α ν B ν + σ ν J ν ) . (3) This equation can be converted to a simpler form dI ν ds = - ( α ν + σ ν )( I ν - S ν ) (4) if we define a source function S ν α ν B ν + σ ν J ν α ν + σ ν , (5) 1 A search in astro-ph, for instance, reveals an ongoing conflict between the Rosseland mean opacity computed by the Opacity Project and that computed by the OPAL group. Why do these people care so intensely about the Rosseland mean opacity? I hope to explain in this lecture. 2 Note that if the specific intensity I ν is isotropic to begin with, coherent scattering doesn’t affect it. 1
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which is just the mean of the Planck function and the angle-averaged specific intensity, weighted by the absorption coefficient and the scattering coefficient, respectively. In general, solving the radiative transfer equation is a tedious chore. How- ever, in studying stellar atmospheres, some simplifying assumptions can be made. The outer layers of a star are relatively low in density, so radiation is the only significant method of transporting energy. 3 Moreover, gradients in temperature and specific intensity are much greater in the radial direction than in the transverse directions. Thus, a stellar atmosphere can usually be well approximated as a plane parallel system, in which properties of the atmosphere, such as T , σ ν , and α ν depend only on the vertical coordinate z . 4 In the plane parallel approximation, the specific intensity I ν depends only on z and on the angle θ between the direction of the light ray and the z axis: when the light travels straight upward, in the same direction that z increases, θ = 0; when the light travels straight down, θ = π . In traversing a thin layer of the atmosphere, of vertical thickness dz , the distance traveled by light is ds = dz cos θ , (6) implying ( ds ) 2 ( dz ) 2 . The radiative transfer equation in a plane parallel atmosphere thus takes the form cos θ ∂I ν ( z, θ ) ∂z = - [ α ν ( z ) + σ ν
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