We derive recurrence relations for the coefficients ak in the power series expansion θ(ξ)=∑ akξ2k of the solution of the Lane—Emden equation, and examine the convergence of these series. For values ...of the polytropic index n<n1≈1.9 the series appear to converge everywhere inside the star. For n>n1 the series converge in the inner part of the star but then diverge. We also derive the series expansions for θ, ξ in powers of m=q2/3, where q=-ξ2dθ/dξ is the polytropic mass. These series appear to converge everywhere within the star for all n ≤ 5. Finally we show that θ(ξ) can be satisfactorily approximated (∼ 1 per cent) by (1-cξ2)/(1+eξ2)m, and give the values of the constants determined by a Padé approximation to the series, and by a two-parameter fit to the numerical solutions.
In order to make asteroseismology a powerful tool to explore stellar interiors, different numerical codes should give the same oscillation frequencies for the same input physics. Any differences ...found when comparing the numerical values of the eigenfrequencies will be an important piece of information regarding the numerical structure of the code. The ESTA group was created to analyze the non-physical sources of these differences. The work presented in this report is a part of Task 2 of the ESTA group. Basically the work is devoted to test, compare and, if needed, optimize the seismic codes used to calculate the eigenfrequencies to be finally compared with observations. The first step in this comparison is presented here. The oscillation codes of nine research groups in the field have been used in this study. The same physics has been imposed for all the codes in order to isolate the non-physical dependence of any possible difference. Two equilibrium models with different grids, 2172 and 4042 mesh points, have been used, and the latter model includes an explicit modelling of semiconvection just outside the convective core. Comparing the results for these two models illustrates the effect of the number of mesh points and their distribution in particularly critical parts of the model, such as the steep composition gradient outside the convective core. A comprehensive study of the frequency differences found for the different codes is given as well. These differences are mainly due to the use of different numerical integration schemes. The number of mesh points and their distribution are crucial for interpreting the results. The use of a second-order integration scheme plus a Richardson extrapolation provides similar results to a fourth-order integration scheme. The proper numerical description of the Brunt-Väisälä frequency in the equilibrium model is also critical for some modes. This influence depends on the set of the eigenfunctions used for the solution of the differential equations. An unexpected result of this study is the high sensitivity of the frequency differences to the inconsistent use of values of the gravitational constant (
G
) in the oscillation codes, within the range of the experimentally determined ones, which differ from the value used to compute the equilibrium model. This effect can provide differences for a given equilibrium model substantially larger than those resulting from the use of different codes or numerical techniques; the actual differences between the values of
G
used by the different codes account for much of the frequency differences found here.
Using ray-tracing techniques, we consider nearly forward refraction in the stellar interior of acoustic waves modified by buoyancy and gravity. Our analysis is based on a local dispersion relation ...developed to second order in the high-frequency asymptotic approximation. The phase shifts δ
ℓ
δ
0(ω)+ℓ(ℓ+1)D(ω) of the partial waves of stellar p modes, which govern the eigenfrequency equation in its classical limit, are developed in terms of explicit integrals containing the radial profiles of seismic parameters in the stellar interior. The accuracy of the resulting description of low-degree stellar p modes is tested using an evolutionary sequence of solar models.
Aims.We show how to construct 2-dimensional models of rapidly rotating stars in hydrostatic equilibrium for any $\Omega(r,\theta)$, given the density $\rho_{\rm m}(r)$ along any one angle ...$\theta_{\rm m}$. If the hydrogen abundance $X_{\rm m}(r)$ is given on $\theta_{\rm m}$ then the adiabatic exponent $\Gamma_1(r,\theta)$ can by determined, yielding a self consistent acoustic model that can be used to investigate the oscillation properties of rapidly rotating stars. Methods.The system of equations governing the hydrostatic structure is solved by iteration using the method of characteristics and spectral expansion, subject to the condition that $\rho(r,\theta)=\rho_{\rm m}(r)$ on $\theta=\theta_{\rm m}$. $\Gamma_1(r,\theta)$ is calculated from the equation of state under the assumption that $X(r,\theta_{\rm m})=X_{\rm m}(r)$ and is constant on surfaces of constant entropy. Alternatively $\Gamma_1$ can be approximated by taking X constant in the equation of state and equal to the surface value. Results.Results are presented for an evolved main sequence star of $2~M_\odot$ with the angular velocity a function only of radius $\Omega=\Omega(r)$, evolved to a central hydrogen abundance of $X_{\rm c}=0.35$. The model is first calculated using a spherically averaged stellar evolution code, where the averaged centrifugal force $2\Omega^2 r/3$ is added to gravity. The resulting $\rho_{\rm m}(r), X_{\rm m}(r)$ are then used as input to determine the 2-dimensional model. Conclusions.The procedure described here gives self consistent hydrostatic and acoustic models of rapidly rotating stars for any $\Omega(r,\theta)$.
We present a new theoretical description of the ‘small frequency separations’ $\delta\omega_{l,n} = \omega_{l,n} - \omega_{l+2,n-1}$ for high-frequency stellar p-modes of low degree, these ...separations being the observable quantities that are primarily sensitive to the structure of the deep stellar interior. The description is based on an integral representation of the phase shift of acoustic waves due to scattering off the stellar core, taking into account the effects of buoyancy and gravitational perturbations. The accuracy of the theoretical description is tested by comparing the predicted frequency separations with values determined by numerically solving the full set of eigenfrequency equations for a standard solar model and for simple zero-age and evolved models of a 3-M⊙ mass main-sequence star with a convective core.
We extend the asymptotic description of solar p-modes to include an excitation source. The linear dynamic response of a star is considered within the general framework of a Fourier transform of the ...source function in time, and a vector spherical harmonic decomposition in space. Quantitative analysis is developed for the linear response to an ‘elementary’ harmonic excitation source described by a δ-function in the radial direction, using a simplified description of the energy leakage from the acoustic cavity. The synthetic p-mode power spectra are computed numerically for different depths of the excitation source, and their simple properties are discussed. The asymptotic eigenfrequency equation is generalized to describe the frequencies of maximum amplitudes in the theoretical response function. The resulting frequency equation with modified ‘surface phase shift’ degenerates into the standard eigenfrequency equation at low frequencies, and describes the high-frequency ‘pseudo-modes’ in the high-frequency limit, joining both these well-known theoretical descriptions in the intermediate frequency range.
PDF
