Mode Mixing and Rotational Splittings: II. Reconciling Different Approaches to Mode Coupling

Abstract: In the mixed-mode asteroseismology of subgiants and red giants, the coupling between the p- and g-mode cavities must be understood well in order to derive localised estimates of interior rotation from measurements of mode multiplet rotational splittings. There exist now two different descriptions of this coupling: one based on an asymptotic quantisation condition, and the other arising from coupling matrices associated with "acoustic molecular orbitals". We examine the analytic properties of both, and derive closed-form expressions for various quantities -- such as the period-stretching function $\tau$ -- which previously had to be solved for numerically. Using these, we reconcile both formulations for the first time, deriving relations by which quantities in each formulation may be translated to and interpreted within the other. This yields an information criterion for whether a given configuration of mixed modes meaningfully constrains the parameters of the asymptotic construction, which is likely not satisfied by the majority of first-ascent red giant stars in our observational sample. Since this construction has been already used to make rotational measurements of such red giants, we examine -- through a hare-and-hounds exercise -- whether, and how, such limitations affect existing measurements. While averaged estimates of core rotation seem fairly robust, template-matching using the asymptotic construction has difficulty reliably assigning rotational splittings to individual multiplets, or estimating mixing fractions $\zeta$ of the most p-dominated mixed modes, where such estimates are most needed. We finally discuss implications for extending the two-zone model of radial differential rotation, e.g. via rotational inversions, with these methods.