Frequency Dependent Ray Paths in Local Helioseismology

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Abstract
The surface of the Sun is continually oscillating due to sound waves encroaching on it from the interior. Measurements of the surface velocity are used to infer some of the properties of the regions through which the sound waves have propagated. Traditionally, this has been done by using a modal decomposition of the surface disturbances. However, the use of ray descriptions, in the form of acoustic holography or time–distance helioseismology, provides an alternative approach which may reveal more detailed information about the properties of local phenomena such as sunspots and active regions. Fundamental to any such treatment is determining the correct ray paths in a given atmosphere. In the simplest approach, the ray paths are constructed to minimise the travel time between two points (Fermat's principle). However, such an approach is only valid in the high frequency limit, ω » ωc, N, where ωc is the acoustic cut-off and N the Brunt-VÄisÄlÄ frequency. Although ωc is often included in time– distance calculations, and N occasionally, the same is not true of acoustic holography. We argue that this raises concerns about image sharpness. As illustrations, representative ray paths are integrated in a realistic solar model to show that the Fermat approximation performs poorly for frequencies of helioseismic interest. We also briefly discuss the importance of the Brunt-VÄisÄlÄ frequency to the time–distance diagram.