I compare values of the frequencies, separation ratios, errors and covariance matrices from a new analysis of 9 solar-like stars with the Legacy project values reported by Lund et al and, for 16Cyg ...A&B and KIC 8379927, with values derived by Davies et al. There is good agreement between my results and Davies’s for these 3 stars, but no such agreement with the Legacy project results. My frequencies differ from the Legacy values, there are inconsistencies in the Legacy frequency covariance matrices which are not positive definite, and the Legacy errors on separation ratios are up to 40 times larger than mine and the values and upper limits derived from the Legacy frequency covariances. There are similar anomalies for 6 other solar-like stars: frequencies and separation ratio errors disagree and 2 have non positive definite covariance matrices. There are inconsistencies in the covariance matrices of 27 the 66 stars in the full Legacy set and problems with the ratio errors for the vast majority of these stars.
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.
I show that given a model star of mass M, radius R, and density profile ρ(x) x = r/R, there exists a two parameter family of models with masses Mk, radii Rk, density profile ρk(x) = λρ(x) and ...frequencies νknℓ = λ1/2νnℓ, where λ,Rk/RA are scaling factors. These models have different internal structures, but all have the same value of separation ratios calculated at given radial orders n, and all exactly satisfy a frequency matching algorithm with an offset function determined as part of the fitting procedure. But they do not satisfy ratio matching at given frequencies nor phase shift matching. This illustrates that erroneous results may be obtained when model fitting with ratios at given n values or frequency matching. I give examples from scaled models and from non scaled evolutionary models.
We critically investigate the practice of adding a power-law surface offset correction f(ν) to the frequencies of stellar models prior to seeking best fit models to an observed star. We show that ...surface layer independent indicators of the internal structure, phase shifts and separation ratios, are displaced in frequency by f(ν) and are therefore not the same as those of the model. Consequently such best fit models do not have exactly the same interior structure as the observed star. Using results on the star HD 177153 we show that the difference between observed and model frequencies for best fit models obtained using surface layer independent procedures have a wide range of different offsets which do not in general follow a Kjeldsen-like power law, and further that best fit models obtained using the offset correction procedure do not necessarily satisfy surface layer independent constraints on the internal structure.
Abstract
Normal-mode oscillation frequencies computed from stellar models differ from those that would be measured from stars with identical interior structures because of modeling errors in the ...near-surface layers. These frequency differences are referred to as the asteroseismic “surface term.” The vast majority of solar-like oscillators that have been observed, and that are expected to be observed in the near future, are evolved stars that exhibit mixed modes. For these evolved stars, the inference of stellar properties from these mode frequencies has been shown to depend on how this surface term is corrected for. We show that existing parameterizations of the surface term account for mode mixing only to first order in perturbation theory, if at all, and therefore may not be adequate for evolved stars. Moreover, existing nonparametric treatments of the surface term do not account for mode mixing. We derive both a first-order construction and a more general approach for one particular class of nonparametric methods. We illustrate the limits of first-order approximations from both analytic considerations and using numerical injection-recovery tests on stellar models. First-order corrections for the surface term are strictly only applicable where the size of the surface term is much smaller than both the coupling strength between the mixed
p
and
g
modes, as well as the local
g
-mode spacing. Our more general matrix construction may be applied to evolved stars, where perturbation theory cannot be relied upon.
Aims. We aim to show that model fitting by searching for a best fit of observed and model separation ratios at the same radial orders n is in principle incorrect, and to show that a correct procedure ...is to compare the model ratios interpolated to the observed frequencies. Methods. We compare models with different interior structures and outer layers, relate the separation ratios to phase shift differences, conduct model fitting experiments using separation ratios, and relate phase shift differences to internal phase shifts. Results. We show that the separation ratios of stellar models with the same interior structure, but different outer layers, are not the same when compared at the same radial order n, but are the same when evaluated at the same frequencies by interpolation. The separation ratios trace the phase shift differences as a function of frequency, not of n, and it is the phase shift differences which are determined by the interior structure. We give examples from model fitting where the ratios at the same n values agree but the models have different interior structure, and where the ratios agree when interpolated to the same frequencies and the models have the same interior structure. The correct procedure is to compare observed ratios with model values interpolated to the observed frequencies.
I discuss several asteroseismology diagnostic techniques that can be applied to the high quality data on stellar oscillations obtained, and to be obtained in the future, from ground based and space ...based experiments. In particular I discuss techniques using the representation of oscillation frequencies in terms of inner and outer phase shifts which can be used both for model fitting and inversion procedures to probe the inner structure of stars, and hence to test and improve our modelling.
The STAROX stellar evolution code Roxburgh, Ian W.
Astrophysics and space science,
08/2008, Letnik:
316, Številka:
1-4
Journal Article
Recenzirano
This paper describes the STAROX stellar evolution code for the calculation of the evolution of a model of a spherical star. The code calculates a model at time
t
k
, that is the run of pressure, ...density, temperature, radius, energy flux and related variables on a mesh in mass
M
i
, given the distribution of chemical elements
X
j
(
i
) at
t
k
and the model at the previous time step
t
k
−1
. It then advances the chemical composition to the next time step
t
k
+1
and calculates a new model at time
t
k
+1
. This process is iterated to convergence. The model equations are solved by Newton–Raphson relaxation; the chemical equations are solved by an iterative procedure, each element being advanced in turn, and the process repeated to convergence. Convection is modelled by a mixing length model and convective mixing is treated as a diffusive process; chemical overshooting can be incorporated in parametric form. The equation of state is taken from OPAL tables and the opacity from a blend of OPAL and Alexander tables. Nuclear reaction rates are from NACRE but only cover the
p
–
p
chain and
CNO
cycle. The atmospheric layers are incorporated in the model by applying the surface boundary condition at small optical depth (
τ
≈0.001). The mesh in mass
M
i
is usually taken as fixed except that there is a moveable mesh point at the boundary of a convective core. Results are given for models of mass 0.9 and 5.0
M
⊙
with initial composition
X
=0.7,
Z
=0.02 evolved to a state where the central hydrogen abundance is
X
c
=0.35, and for a model of mass 2.0
M
⊙
with initial
X
=0.72,
Z
=0.02, evolved to
X
c
=0.01 and with core overshooting. In this latter case we compute two models one with and one without a moveable mesh point at the boundary of the convective core to illustrate the importance of having such a moveable mesh point for the determination of the Brunt–Väisälä frequency in the layers outside the core.
The advent of space-based photometry missions in the early 21st century enabled the application to asteroseismic data of advanced inference techniques until then restricted to the field of ...helioseismology. The high quality of the observations, the discovery of mixed modes in evolved solar-like oscillators and the need for an improvement in the determination of stellar fundamental parameters such as mass, radius and age led to the development of sophisticated modelling tools, amongst which seismic inversions play a key role. In this review, we will discuss the existing inversion techniques for the internal structure of distant stars adapted from helio-to asteroseismology. We will present results obtained for various Kepler targets, their coupling to other existing modelling techniques as well as the limitations of seismic analyses and the perspectives for future developments of these approaches in the context of the current TESS and the future PLATO mission, as well as the exploitation of the mixed modes observed in post-main sequence solar-like oscillators, for which variational formulations might not provide sufficient accuracy.
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