Surface layer independent model fitting by phase matching: theory and application to HD 49933 and HD 177153 (aka Perky)

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

Received: 5 November 2014
Accepted: 15 December 2014

Abstract

Aims. Our aim is to describe the theory of surface layer independent model fitting by phase matching and to apply this to the stars HD 49933 observed by CoRoT, and HD 177153 (aka Perky) observed by Kepler.

Methods. We use theoretical analysis, phase shifts, and model fitting.

Results. We define the inner and outer phase shifts of a frequency set of a model star and show that the outer phase shifts are (almost) independent of degree ℓ, and that a function of the inner phase shifts (the phase function) collapses to an ℓ independent function of frequency in the outer layers. We then show how to use this result in a model fitting technique to find a best fit model to an observed frequency set by calculating the inner phase shifts of a model using the observed frequencies and determining the extent to which the phase function collapses to a single function of frequency in the outer layers. This technique does not depend on the radial order n assigned to the observed frequencies. We give two examples applying this technique to the frequency sets of HD 49933 observed by CoRoT and HD 177153 (aka Perky) observed by Kepler, for which measurements of angular diameters and bolometric fluxes are available. For HD 49933 we find a very wide range of models to be consistent with the data (all with convective core overshooting) – and conclude that the data is not precise enough to make any useful restrictions on the structure of this star. For HD 177153 our best fit models have no convective cores, masses in the range 1.15−1.17 M_⊙, ages of 4.45−4.70 × 10⁹ yr, Z in the range 0.021−0.024, XH = 0.71−0.72, Y = 0.256 − 0.266 and mixing length parameter α = 1.8. We compare our results to those of previous studies. We contrast the phase matching technique to that using the ratios of small to large separations, showing that it avoids the problem of correlated errors in separation ratio fitting and of assigning radial order n to the modes.