Semianalytic Expressions for the Isolation and Coupling of Mixed Modes

In the oscillation spectra of giant stars, nonradial modes may be seen to undergo avoided crossings, which produce a characteristic "mode bumping" of the otherwise uniform asymptotic p- and g-mode patterns in their respective echelle diagrams. Avoided crossings evolve very quickly relative to typical observational errors and are therefore extremely useful in determining precise ages of stars, particularly in subgiants. This phenomenon is caused by coupling between modes in the p- and g-mode cavities that are near resonance with each other. Most theoretical analyses of the coupling between these mode cavities rely on the Jeffreys-Wentzel-Kramers-Brillouin approach, which is strictly speaking inapplicable to the low-order g modes observed in subgiants or the low-order p modes seen in very evolved red giants. We present both a nonasymptotic prescription for isolating the two mode cavities, as well as a perturbative (and also nonasymptotic) description of the coupling between them, which we show to hold good for the low-order g and p modes in these physical situations. Finally, we discuss how these results may be applied to modeling subgiant stars and determining their global properties from oscillation frequencies. We also make our code for all of these computations publicly available.