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We describe a general statistical perspective for the study of radiative transfer through inhomogeneous media and apply it to simple stochastic atmospheric models. The particular context for our applications considers a stochastic atmosphere to be a multi-component medium in which any individual component of the medium is locally smooth. The stochastic nature of the atmosphere resides in the statistical character of the complex network of boundaries that separate various species of media one from another. We illustrate the theory with simple atmospheric models based on an ambient medium into which are randomly embedded structural elements containing alternative species of medium. We consider structures of various shapes and sizes, ranging from simple spheres to elongated or fluted structures with preferred orientation.
An important distinctive quality of a stochastic atmosphere is whether the medium contains structures that individually may be optically thick. Atmospheres containing only optically thin structures tend to be statistically amenable to representation by equivalent smooth atmospheres. The theory we have developed is fully applicable to atmospheres that contain optically thick elements as well as optically thin ones. Such conditions apply to a broad variety of radiative transfer problems in astrophysics and stellar physics, for example, to emission from interstellar gas clouds, from solar or stellar chromospheres or from photospheres that contain heated magnetic flux tubes.
In this work we concentrate on a formalism that rests on the Markov assumption, which states that the probability of encountering a transition from one type of medium, $A$, to another, $B$, is independent of the cumulative distance since the transition into medium $A$, as one proceeds along the optical path. We examine the importance of this assumption and its utility as a first approximation by illustrating the consequences of its application to atmospheric models that are non-Markovian.
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