Structural and Evolutionary Diagnostics from Asteroseismic Phase Functions

In the asymptotic parameterisation of mode frequencies, the phase function $\epsilon(\nu)$ completely specifies the detailed structure of the frequency eigenvalues. In practice, however, this function of frequency is reduced to a single scalar $\epsilon$, defined, particularly by observers, as the intercept of a least-squares fit to the frequencies against radial order, or via the central value of this function. The procedure by which this is done is not unique. We derive a few simple expressions relating various observational estimators of $\epsilon$ for radial modes to each other, and to the underlying theoretical object. In particular we demonstrate that a ``reduced' functional parameterisation is both insensitive to mis-estimations of $\Delta\nu$, and easy to evaluate locally in terms of both observational and theoretical quantities. It has been shown previously that such a local definition of $\epsilon$ can distinguish between stars on the ascending part of the red giant branch and those in the red clump. We find that this sensitivity to evolutionary stage arises from differences in the local frequency derivative of the underlying phase function, a consequence of differences in internal structure. By constructing an HR-like diagram out of purely seismic observables, we provide a unified view of the \textit{Kepler} asteroseismic sample, as well as the initial results from \textit{TESS}. We investigate how various astrophysical quantities and modelling parameters affect the morphology of isochrones on this seismic diagram. We also show that $\epsilon$ can be used as an independent input when deriving stellar parameters from global asteroseismic quantities.