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

## TL;DR

This paper compares two theoretical approaches to mode coupling in mixed-mode asteroseismology, derives relations to reconcile them, and assesses how these impact measurements of stellar interior rotation.

## Contribution

It provides the first formal reconciliation of asymptotic and matrix-based mode coupling descriptions, enabling better interpretation of mixed-mode observations.

## Key findings

- Reconciled the asymptotic and matrix formulations of mode coupling.
- Derived closed-form expressions for key quantities like the period-stretching function.
- Identified limitations in current methods for estimating mode parameters in red giants.

## 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.