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

**Authors:** J. M. Joel Ong, Charlotte Gehan  
**arXiv:** [2302.12402](https://arxiv.org/abs/2302.12402)  
**Date:** 2023-02-24  
**Summary page:** https://www.summarizepaper.com/en/paper/2302.12402v1/

## Key points

- The paper focuses on the coupling between p- and g-mode cavities in mixed-mode asteroseismology of subgiants and red giants.
- Understanding this coupling is crucial for deriving localized estimates of interior rotation from measurements of mode multiplet rotational splittings.
- The authors examine the analytic properties of both descriptions and derive closed-form expressions for various quantities that previously had to be solved numerically.
- Using these closed-form expressions, the authors 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 their observational sample.
- The authors also examine how such limitations affect existing measurements through a hare-and-hounds exercise.
- They find that 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.
- Finally, this paper discusses implications for extending the two-zone model of radial differential rotation via rotational inversions with these methods.
- The closed-form nature of the asymptotic parameterization makes it far easier to operationalize and computationally less expensive than matrix decompositions employed in algebraic approaches; numerical solutions to its eigenvalue problem are not required either.
- Overall, this study provides a comprehensive analysis and reconciliation between two different descriptions used in mixed-mode asteroseismology while highlighting their limitations when applied to observational data.

## Summary

This paper focuses on the coupling between p- and g-mode cavities in mixed-mode asteroseismology of subgiants and red giants. The understanding of this coupling is crucial for deriving localized estimates of interior rotation from measurements of mode multiplet rotational splittings. This paper examines the analytic properties of both descriptions and derives closed-form expressions for various quantities that previously had to be solved numerically. Using these closed-form expressions, the authors 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 their observational sample. The authors also examine how such limitations affect existing measurements through a hare-and-hounds exercise. They find that 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. Finally, this paper discusses implications for extending the two-zone model of radial differential rotation via rotational inversions with these methods. The closed-form nature of the asymptotic parameterization makes it far easier to operationalize and computationally less expensive than matrix decompositions employed in algebraic approaches; numerical solutions to its eigenvalue problem are not required either. Overall, this study provides a comprehensive analysis and reconciliation between two different descriptions used in mixed-mode asteroseismology while highlighting their limitations when applied to observational data.