Asteroseismic model fitting by comparing ϵ_nℓ values

Astronomy Unit, Queen Mary University of London, Mile End Road, London E1 4NS, UK
e-mail: I.W.Roxburgh@qmul.ac.uk

Received: 25 May 2015
Accepted: 26 October 2015

Abstract

We present an asteroseismic model fitting algorithm based on comparing model and observed ϵ_ℓ(ν) values defined in terms of frequencies by ν_nℓ = Δ [ n + ℓ/ 2 + ϵ_ℓ(ν_nℓ) ] where Δ is an average large separation. We show that if two stellar models have the same interior structure but different outer layers then the difference between their ϵ_ℓ(ν) values, interpolated to the same frequencies, collapses to a function only of frequency, independent of angular degree ℓ. The algorithm tests the goodness fit by comparing the difference in model and observed ϵ values after having subtracted off a best fit ℓ independent function of frequency ℱ(ν), and only requires interpolation in model values and not in observed values so the errors on the observed values are uncorrelated; it is independent of the n values assigned to the radial ordering of the frequencies and does not require the calculation of inner phase shifts of the model. We contrast this to a proposed direct frequency matching technique which minimises the difference between observed and model frequencies after having subtracted off an ℓ independent fit to these differences. We show this technique is flawed in principle, that all models with the same dimensionless structure but any mass and radius have the same quality of fit to an observed data set, and that it can give erroneous best fit models. We illustrate the epsilon matching technique by comparing stellar models and then apply it to data on HD 177153 (aka Perky). On comparing observations with a set of main sequence evolutionary models we find that models which satisfy constraints on the luminosity, radius, Δ, and on ϵ matching, have masses in the range 1.155 ± 0.035 M_⊙ and ages in the range 4.486 ± 0.250 × 10⁹ yr. Since the large separation and the radius are surface layer dependent we examine “pure surface layer independent” model fitting where the only constraints on the model fitting are on the luminosity and epsilon matching, and show that the best fit models have M/M_⊙ = 1.13 ± 0.06 and age = 4.62 ± 0.39 × 10⁹ yr.